Download Algebraic Quantum Field Theory on Curved Spacetimes

Chapter 2
Algebraic Quantum Field Theory
on Curved Spacetimes
Abstract In this chapter, we review all background material on algebraic quantum
field theory on curved spacetimes which is necessary for understanding the cosmological applications discussed in the next chapter. Starting with a brief account of
globally hyperbolic curved spacetimes and related geometric notions, we then explain
how the algebras of observables generated by products of linear quantum fields at
different points are obtained by canonical quantization of spaces of classical observables. This discussion will be model-independent and will cover both Bosonic and
Fermionic models with and without local gauge symmetries. Afterwards, we review
the concept of Hadamard states which encompass all physically reasonable quantum
states on curved spacetimes. The modern paradigm in QFT on curved spacetimes
is that observables and their algebras should be constructed in a local and covariant
way. We briefly review the theoretical formulation of this concept and explain how
it is implemented in the construction of an extended algebra of observables of the
free scalar field which also contains products of quantum fields at coinciding points.
Finally, we discuss the quantum stress-energy tensor as a particular example of such
an observable as well as the related semiclassical Einstein equation.
2.1 Globally Hyperbolic Spacetimes and Related
Geometric Notions
The philosophy of algebraic quantum field theory in curved spacetimes is to set up
a framework which is valid on all physically reasonable curved Lorentzian spacetimes and independent of their particular properties. Given this framework, one may
then exploit particular properties of a given spacetime such as symmetries in order
obtain specific results or to perform explicit calculations. A class of spacetimes
which encompasses most cases which are of physical interest are globally hyperbolic
spacetimes. These include Friedmann-Lemaître-Robertson-Walker spacetimes-in
particular Minkowski spacetime-as well as Black Hole spacetimes such as Schwarzschild-and Kerr-spacetime, whereas prominent examples of spacetimes which are not
globally hyperbolic are Anti de Sitter-spacetime (see e.g. [5, Chap. 3.5]) and a portion
of Minkowski spacetime obtained by restricting one of the spatial coordinates to a
© The Author(s) 2016
T.-P. Hack, Cosmological Applications of Algebraic Quantum Field Theory
in Curved Spacetimes, SpringerBriefs in Mathematical Physics 6,
DOI 10.1007/978-3-319-21894-6_2
13
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2 Algebraic Quantum Field Theory on Curved Spacetimes
finite interval such as the spacetimes relevant for discussing the Casimir effect. The
constructions we shall review in the following are well-defined on all globally hyperbolic spacetimes. The physically relevant spacetime examples which are not globally
hyperbolic are usually such that sufficiently small portions still have this property.
Consequently, the algebraic constructions on globally hyperbolic spacetimes can be
extended to these cases by patching together local constructions, see for instance [33,
85]. In this section we shall review the definition of globally hyperbolic spacetimes
and a few related differential geometric notions which we shall use throughout this
monograph.
To this end, in this work a spacetime (M, g) is meant to be a Hausdorff, connected, smooth manifold M, endowed with a Lorentzian
metric g, the invariant
. √
volume measure of which shall be denoted by dg x = | det g|d x. We will mostly
consider four-dimensional spacetimes. However, most notions and results can be
formulated and obtained for Lorentzian spacetimes with a dimension d differing
from four and we will try to point out how the spacetime dimension affects them
whenever it seems interesting and possible. We will follow the monograph by Wald
[122] regarding most conventions and notations and, hence, work with the metric
signature (−, +, +, +). It is often required that a spacetime be second countable,
or, equivalently, paracompact, i.e. that its topology has a countable basis. Though,
as proven by Geroch in [60], paracompactness already follows from the properties
of (M, g) listed above. In addition to the attributes already required, we demand
that the spacetime under consideration is orientable and time-orientable and that an
orientation has been chosen in both respects. We will often omit the spacetime metric
g and denote a spacetime by M in brief.
For a point x ∈ M, Tx M denotes the tangent space of M at x and Tx∗ M denotes the
respective cotangent space; the tangent and cotangent bundles of M shall be denoted
by T M and T ∗ M, respectively. If χ : M1 → M2 is a diffeomorphism, we denote by
χ ∗ the pull-back of χ and by χ∗ the push-forward of χ . χ ∗ and χ∗ map tensors on M2
to tensors on M1 and tensors on M1 to tensors on M2 , respectively; they furthermore
satisfy χ∗ = (χ −1 )∗ [122, Appendix C]. In case g1 and g2 are the chosen Lorentzian
metrics on M1 and M2 and χ∗ g1 = g2 , we call χ an isometry; if χ∗ g1 = Ω 2 g2 with a
strictly positive smooth function Ω, χ shall be called a conformal isometry and Ω 2 g
a conformal transformation of g. Note that this definition differs from the one often
used in the case of highly symmetric or flat spacetimes since one does not rescale
coordinates, but the metric. A conformal transformation according to our definition
is sometimes called Weyl transformation in the literature. If χ is an embedding χ :
M1 → M2 , i.e. χ (M1 ) is a submanifold of M2 and χ a diffeomorphism between M1
and χ (M1 ), it is understood that a push-forward χ∗ of χ is only defined on χ (M1 ) ⊂
M2 . In case an embedding χ : M1 → M2 between the manifolds of two spacetimes
(M1 , g1 ) and (M2 , g2 ) is an isometry between (M1 , g1 ) and (χ (M1 ), g2 |χ (M1 ) ), we
call χ an isometric embedding, whereas an embedding which is a conformal isometry
between (M1 , g1 ) and (χ (M1 ), g2 |χ (M1 ) ) shall be called a conformal embedding.
Some works make extensive use of the abstract index notation, i.e. they use Latin
indices to denote tensorial identities which hold in any basis to distinguish them
from identities which hold only in specific bases. As this distinction will not be
2.1 Globally Hyperbolic Spacetimes and Related Geometric Notions
15
necessary in the present work, we will not use abstract index notation, but shall use
Greek indices to denote general tensor components in a coordinate basis {∂μ }μ=0,...,3
and shall reserve Latin indices
for other uses. We employ the Einstein summation
μ . μ
convention, e.g. Aμ = 3μ=0 Aμ , and we shall lower Greek indices by means of
.
.
gμν = g(∂μ , ∂ν ) and raise them by g μν = (g −1 )μν .
Every smooth Lorentzian manifold admits a unique metric-compatible and torsionfree linear connection, the Levi-Civita connection, and we shall denote the associated covariant derivative along a vector field v, i.e. a smooth section of T M,
by ∇v . We will abbreviate ∇∂μ by ∇μ and furthermore use the shorthand notation
.
T;μ1 ···μn = ∇μ1 · · · ∇μn T for covariant derivatives of a tensor field T . Our definitions
for the Riemann tensor Rαβγ δ , the Ricci tensor Rαβ , and the Ricci scalar R are
.
vα;βγ − vα;γβ = Rα λβγ vλ ,
.
Rαβ = Rα λβλ ,
.
R = R αα ,
(2.1)
where vα are the components of an arbitrary covector. The Riemann tensor possesses
the symmetries
Rαβγ δ = −Rβαγ δ = Rγ δαβ ,
Rαβγ δ + Rαδβγ + Rαγ δβ = 0
(2.2)
and fulfils the Bianchi identity
Rαβγ δ;ε + Rαβεγ ;δ + Rαβδε;γ = 0 .
(2.3)
Moreover, its trace-free part, the Weyl tensor, is defined as
Cαβγ δ = Rαβγ δ −
−
1
gαδ gβγ − gαγ gβδ R
6
1
gβδ Rαγ − gβγ Rαδ − gαδ Rβγ + gαγ Rβδ ,
2
where the appearing coefficients differ in spacetimes with d = 4. In addition to the
covariant derivative, we can define the notion of a Lie derivative along a vector field
v: the integral curves c(s) of v with respect to a curve parameter s define, in general
only for small s and on an open neighbourhood of c(0), a one-parameter group of
diffeomorphisms χsv [122, Chap. 2.2]. Given a tensor field T of arbitrary rank, we
can thus define the Lie derivative of T along v as
.
Lv T = lim
s→0
v ∗
(χ−s
) T −T
s
.
If χsv is a one-parameter group of isometries, we call v a Killing vector field, while
in case of χsv being a one-parameter group of conformal isometries, we shall call v a
conformal Killing vector field. It follows that a Killing vector field v fulfils Lv g = 0,
while a conformal Killing vector field v fulfils Lv g = f g with some smooth function
f [122, Appendix C.3].
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2 Algebraic Quantum Field Theory on Curved Spacetimes
In order to define what it means for a spacetime to be globally hyperbolic, we need
a few additional standard notions related to Lorentzian spacetimes. To wit, following
our sign convention, we call a vector vx ∈ Tx M timelike if g(vx , vx ) < 0, spacelike
if g(vx , vx ) > 0, lightlike or null if g(vx , vx ) = 0, and causal if it is either timelike
or null. Extending this, we call a vector field v : M → T M spacelike, timelike,
lightlike, or causal if it possesses this property at every point. Finally, we call a curve
c : R ⊃ I → M, with I an interval, spacelike, timelike, lightlike, or causal if its
tangent vector field bears this property. Note that, according to our definition, a trivial
curve c ≡ x is lightlike. As (M, g) is time orientable, we can split the lightcones
in T M at all points in M into ‘future’ and ‘past’ in a consistent way and say that a
causal curve is future directed if its tangent vector field at a point is always in the
future lightcone at this point; past directed causal curves are defined analogously.
For the definition of global hyperbolicity, we need the notion of inextendible
causal curves; these are curves that ‘run off to infinity’ or ‘run into a singular point’.
Hence, given a future directed curve c parametrised by s, we call x a future endpoint
of c if, for every neighbourhood O of x, there is an s0 such that c(s) ∈ O for all s > s0 .
With this in mind, we say that a future directed causal curve is future inextendible
if, for all possible parametrisations, it has no future endpoint and we define past
inextendible past directed causal curves similarly. A related notion is the one of a
complete geodesic. A geodesic c is called complete if, in its affine parametrisation
defined by ∇dc/ds dc
ds = 0, the affine parameter s ranges over all R. A manifold M is
called geodesically complete if all geodesics on M are complete.
In the following, we are going to define the generalisations of flat spacetime
lightcones in curved spacetimes. By I + (x, M) we denote the chronological future of
a point x relative to M, i.e. all points in M which can be reached by a future directed
timelike curve starting from x, while J + (x, M) denotes the causal future of a point
x, viz. all points in M which can be reached by future directed causal curve starting
from x. Notice that, generally, x ∈ J + (x, M) and I + (x, M) is an open subset of M
while the situations x ∈
/ I + (x, M) and J + (x, M) being a closed subset of M are not
generic, but for instance present in globally hyperbolic spacetimes [122]. In analogy
to the preceding definitions, we define the chronological past I − (x, M) and causal
past J − (x, M) of a point x by employing past directed timelike and causal curves,
respectively. We extend this definition to a general subset O ⊂ M by setting
. ±
I (x, M)
I ± (O, M) =
x∈O
. ±
J ± (O, M) =
J (x, M) ;
x∈O
.
.
additionally, we define I (O, M) = I + (O, M) ∪ I − (O, M) and J (O, M) =
+
−
J (O, M) ∪ J (O, M). As the penultimate prerequisite for the definition of global
hyperbolicity, we say that a subset O of M is achronal if I + (O, M) ∩ O is empty,
i.e. an achronal set is such that every timelike curve meets it at most once. Given
a closed achronal set O, we define its future domain of dependence D + (O, M) as
the set containing all points x ∈ M such that every past inextendible causal curve
through x intersects O. By our definitions, D + (O, M) ⊂ J + (O, M), but note that
2.1 Globally Hyperbolic Spacetimes and Related Geometric Notions
17
J + (O, M) is in general considerably larger than D + (O, M). We define D − (O, M)
.
analogously and set D(O, M) = D + (O, M) ∪ D − (O, M). D(O, M) is sometimes
also called the Cauchy development of O. With this, we are finally in the position to
state the definition of global hyperbolicity (valid for all spacetime dimensions).
Definition 2.1 A Cauchy surface is a closed achronal set Σ ⊂ M with D(Σ, M) =
M. A spacetime (M, g) is called globally hyperbolic if it contains a Cauchy surface.
Although the geometric intuition sourced by our knowledge of Minkowski spacetime can fail us in general Lorentzian spacetimes, it is essentially satisfactory in
globally hyperbolic spacetimes. According to Definition 2.1, a Cauchy surface is
a ‘non-timelike’ set on which every ‘physical signal’ or ‘worldline’ must register
exactly once. This is reminiscent of a constant time surface in flat spacetime and one
can indeed show that this is correct. In fact, Geroch has proved in [61] that globally
hyperbolic spacetimes are topologically R × Σ and Bernal and Sanchez [9–11] have
been able to improve on this and to show that every globally hyperbolic spacetime
has a smooth Cauchy surface Σ and is, hence, even diffeomorphic to R × Σ. This
implies in particular the existence of a (non-unique) smooth global time function
t : M → R, i.e. t is a smooth function with a timelike and future directed gradient
field ∇t; t is, hence, strictly increasing along any future directed timelike curve. In
the following, we shall always consider smooth Cauchy surfaces, even in the cases
where we do not mention it explicitly.
In the remainder of this chapter, we will gradually see that globally hyperbolic
curved spacetimes have many more nice properties well-known from flat spacetime
and, hence, seem to constitute the perfect compromise between a spacetime which is
generically curved and one which is physically sensible. Particularly, it will turn out
that second order, linear, hyperbolic partial differential equations have well-defined
global solutions on a globally hyperbolic spacetime. Hence, whenever we speak of a
spacetime in the following and do not explicitly demand it to be globally hyperbolic,
this property shall be understood to be present implicitly.
On globally hyperbolic spacetimes, there can be no closed timelike curves, otherwise we would have a contradiction to the existence of a smooth and strictly
increasing time function. There is a causality condition related to this which can be
shown to be weaker than global hyperbolicity, namely, strong causality. A spacetime
is called strongly causal if it can not contain almost closed timelike curves, i.e. for
every x ∈ M and every neighbourhood O1 x, there is a neighbourhood O2 ⊂ O1
of x such that no causal curve intersects O2 more than once. One might wonder if this
weaker condition can be filled up to obtain full global hyperbolicity and indeed some
references, e.g. [5, 66], define a spacetime (M, g) to be globally hyperbolic if it is
strongly causal and J + (x) ∩ J − (y) is compact for all x, y ∈ M. One can show that
the latter definition is equivalent to Definition 2.1 [5, 122] which is, notwithstanding,
the more intuitive one in our opinion.
We close this section by introducing a few additional sets with special causal
properties. To this avail, we denote by expx the exponential map at x ∈ M. A
set O ⊂ M is called geodesically starshaped with respect to x ∈ O if there is
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2 Algebraic Quantum Field Theory on Curved Spacetimes
an open subset O of Tx M which is starshaped with respect to 0 ∈ Tx M such
that expx : O → O is a diffeomorphism. We call a subset O ⊂ M geodesically
convex if it is geodesically starshaped with respect to all its points. This entails in
particular that each two points x, y in O are connected by a unique geodesic which is
completely contained in O. A related notion are causal domains, these are subsets of
geodesically convex sets which are in addition globally hyperbolic. Finally, we would
like to introduce causally convex regions, a generalisation of geodesically convex sets.
They are open, non-empty subsets O ⊂ M with the property that, for all x, y ∈ O,
all causal curves connecting x and y are entirely contained in O. One can prove that
every point in a spacetime lies in a geodesically convex neighbourhood and in a causal
domain [57] and one might wonder if the case of a globally hyperbolic spacetime
which is geodesically convex is not quite generic. However, whereas FriedmannLemaître–Robertson-Walker spacetimes with flat spatial sections are geodesically
convex, even de Sitter spacetime, which is both globally hyperbolic and maximally
symmetric and could, hence, be expected to share many properties of Minkowski
spacetime, is not.
2.2 Linear Classical Fields on Curved Spacetimes
As outlined in Sect. 1.1, the ‘canonical’ route to quantize linear classical field
theories on curved spacetimes in the algebraic language is to first construct the
canonical covariant classical Poisson bracket (or a symmetric equivalent in the
case of Fermionic theories) and then to quantize the model by enforcing canonical (anti)commutation relations defined by this bracket. In this section, we shall
first review how this is done for free field theories without gauge symmetry before
discussing the case where local gauge symmetries are present.
2.2.1 Models Without Gauge Symmetry
We shall start our discussion of classical field theories without local gauge symmetries
by looking at the example of the free Klein-Gordon field, which is the ‘harmonic
oscillator’ of QFT on curved spacetimes. In discussing this example it will become
clear what the basic ingredients determining a linear field theoretic model are and how
they enter the definition and construction of this model in the algebraic framework.
2.2.1.1 The Free Neutral Klein-Gordon Field
In Physics, we are used to describe dynamics by (partial) differential equations and
initial conditions. The relevant equation for the neutral scalar field is the free KleinGordon equation
2.2 Linear Classical Fields on Curved Spacetimes
19
.
(− + f )φ = (−∇μ ∇ μ + f )φ = 0
with the d’Alembert operator and some scalar function f of mass dimension 2.
The function f determines the ‘potential’ V (φ) = 21 f φ 2 of the Klein-Gordon field
and may be considered as a background field just like the metric g. Usually one
considers the case f = m 2 + ξ R, i.e.
.
Pφ = − + ξ R + m 2 φ = 0
(2.4)
such that f is entirely determined in terms of a constant mass m ≥ 0 and the Ricci
scalar R where the dimensionless constant ξ parametrises the strength of the coupling
of φ to R. In principle one could consider more non-trivial coupling terms with the
correct mass dimension such as f = (R αβ Rαβ )2 /(m 4 R), however these may be
ruled out by either invoking Occam’s razor or by demanding analytic dependence of
f on the metric and m like in [70, Sect. 5.1].
The case (2.4) with ξ = 0 is usually called minimal coupling, whereas, in four
dimensions, the case ξ = 1/6 is called conformal coupling. While the former name
refers to the fact that the Klein-Gordon field is coupled to the background metric only
via the covariant derivative, the reason for the latter is rooted in the behaviour of this
derivative under conformal transformations. Namely, if we consider the conformally
.
related metrics g and g = Ω 2 g with a strictly positive smooth function Ω, denote
and R
the quantities
,
by ∇, , and R the quantities associated to g and by ∇,
associated to g , then the respective metric compatibility of the covariant derivatives
and their agreement on scalar functions imply [122, Appendix D]
∇ and ∇
+
−
1
1
1
1
= 3 − + R .
R
6
Ω
Ω
6
(2.5)
This entails that a function φ solving (− + 16 R)φ = 0 can be mapped to a solution
+ 1 R)
φ
of (−
= 0 by multiplying it with the conformal factor Ω to the power
φ
6
= Ω −1 φ. We shall therefore call a scalar field
of the conformal weight −1, i.e. φ
φ with an equation of motion (− + 16 R)φ = 0 conformally invariant. In other
spacetimes dimensions d = 4, the conformal weight and the magnitude of the
conformal coupling are different, see [122, Appendix D].
Having a partial differential equation for a free scalar field at hand, one would
expect that giving sufficient initial data would determine a unique solution on all M.
However, this is, in case of the Klein-Gordon operator at hand, in general only true
for globally hyperbolic spacetimes. To see a simple counterexample, let us consider
Minkowski spacetime with a compactified time direction and the massless case,
i.e. the equation (∂t2 − ∂x2 − ∂ y2 − ∂z2 )φ = 0. Giving initial conditions φ|t=0 = 0,
∂t φ|t=0 = 1, a possible local solution is φ ≡ t. But this can of course never be a
global solution, since one would run into contradictions after a full revolution around
the compactified time direction.
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2 Algebraic Quantum Field Theory on Curved Spacetimes
In what follows, the fundamental solutions or Green’s functions of the KleinGordon equation shall play a distinguished role. Before stating their existence, as
well as the existence of general solutions, let us define the function spaces we shall
be working with in the following, as well as their topological duals, see e.g. [22,
Chap. VI] for an introduction.
.
Definition 2.2 By Γ (M) = C ∞ (M, R) we denote the smooth (infinitely often continuously differentiable), real-valued functions on M equipped with the usual locally
convex topology, i.e. a sequence of functions f n ∈ Γ (M) is said to converge to
f ∈ Γ (M) if all derivatives of f n converge to the ones of f uniformly on all compact subsets of M.
.
The space Γ0 (M) = C0∞ (M, R) is the subset of Γ (M) constituted by the smooth,
real-valued functions with compact support. We equip Γ0 (M) with the locally convex
topology determined by saying that a sequence of functions f n ∈ Γ0 (M) converges
to f ∈ Γ0 (M) if there is a compact subset K ⊂ M such that all f n and f are
supported in K and all derivatives of f n converge to the ones of f uniformly in K .
.
.
By Γ0C (M) = Γ0 (M) ⊗R C, Γ C (M) = Γ (M) ⊗R C we denote the complexifications of Γ0 (M) and Γ (M) respectively.
The spaces Γsc (M) and Γtc (M) denote the subspaces of Γ (M) consisting of
functions with spacelike-compact and timelike-compact support respectively. I.e.
supp f ∩ Σ is compact for all Cauchy surfaces Σ of (M, g) and all f ∈ Γsc (M),
whereas for all f ∈ Γtc (M) there exist two Cauchy surfaces Σ1 , Σ2 with supp f ⊂
J − (Σ1 , M) ∩ J + (Σ2 , M).
By Γ0 (M) we denote the space of distributions, i.e. the topological dual of Γ0 (M)
provided by continuous, linear functionals Γ0 (M) → R, whereas Γ (M) denotes
the topological dual of Γ (M), i.e. the space of distributions with compact support.
Γ C (M) and Γ0C (M) denote the complexified versions of the real-valued spaces.
For f ∈ Γ (M) and u ∈ Γ0 (M) ⊃ Γ (M) ⊃ Γ0 (M) with compact overlapping
support, we shall denote the (symmetric and non-degenerate) dual pairing of f and
u by
.
u, f = dg x u(x) f (x) .
M
The physical relevance of the above spaces is that functions in Γ0 (M), so-called
test functions, should henceforth essentially be viewed as encoding the localisation
of some observable in space and time, reflecting the fact that a detector is of finite
spatial extent and a measurement is made in a finite time interval. From the point
of view of dynamics, initial data for a partial differential equation may be encoded
by distributions or functions with both compact and non-compact support, whereas
solutions of hyperbolic partial differential equations like the Klein-Gordon one are
typically distributions or smooth functions which do not have compact support on
account of the causal propagation of initial data; having a solution with compact
support in time would entail that data ‘is lost somewhere’. Moreover, fundamental
solutions of differential equations will always be singular distributions, as can be
expected from the fact that they are solutions with a singular δ-distribution as source.
2.2 Linear Classical Fields on Curved Spacetimes
21
Finally, since (anti)commutation relations of quantum fields are usually formulated
in terms of fundamental solutions, the quantum fields and their expectation values
will also turn out to be singular distributions quite generically. Physically this stems
from the fact that a quantum field has infinitely many degrees of freedom.
Let us now state the theorem which guarantees us existence and properties of
solutions and fundamental solutions (also termed Green’s functions or propagators)
of the Klein-Gordon operator P. We refer to the monograph [5] for the proofs.
Theorem 2.1 Let P : Γ (M) → Γ (M) be a normally hyperbolic operator on a
globally hyperbolic spacetime (M, g), i.e. in each coordinate patch of M, P can be
expressed as
P = −g μν ∂μ ∂ν + Aμ ∂μ + B
with smooth functions Aμ , B and the metric principal symbol −g μν ∂μ ∂ν . Then, the
following results hold.
1. Let f ∈ Γ0 (M), let Σ be a smooth Cauchy surface of M, let (u 0 , u 1 ) ∈ Γ0 (Σ) ×
Γ0 (Σ), and let N be the future directed timelike unit normal vector field of Σ.
Then, the Cauchy problem
Pu = f,
u|Σ ≡ u 0 ,
∇ N u|Σ ≡ u 1
has a unique solution u ∈ Γ (M). Moreover,
supp u ⊂ J (supp f ∪ supp u 0 ∪ supp u 1 , M) .
A unique solution to the Cauchy problem also exists if the assumptions on the
compact support of f , u 0 and u 1 are dropped.
2. There exist unique retarded E R and advanced E A fundamental solutions (Green’s
functions, propagators) of P. Namely, there are unique continuous maps E R/A :
Γ0 (M) → Γ (M) satisfying P ◦ E R/A = E R/A ◦ P = idΓ0 (M) and supp E R/A f ⊂
J ± (supp f, M) for all f ∈ Γ0 (M).
3. Let f , g ∈ Γ0 (M). If P is formally selfadjoint, i.e. f, Pg = P f, g, then E R
and E A are the formal adjoints of one another, namely, f, E R/A g = E A/R f, g.
.
4. The causal propagator (Pauli-Jordan function) of P defined as E = E R − E A is
a continuous map Γ0 (M) → Γsc (M) ⊂ Γ (M) satisfying: for all solutions u of
Pu = 0 with compactly supported initial conditions on a Cauchy surface there
is an f ∈ Γ0 (M) such that u = E f . Moreover, for every f ∈ Γ0 (M) satisfying
E f = 0 there is a g ∈ Γ0 (M) such that f = Pg. Finally if P is formally
self-adjoint, then E is formally skew-adjoint, i.e. f, Eg = −E f, g .
The Klein-Gordon operator P is manifestly normally hyperbolic. Moreover, one can
check by partial integration that P is also formally self-adjoint. Hence, all abovementioned results hold for P.
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2 Algebraic Quantum Field Theory on Curved Spacetimes
By continuity and the fact that Γ (M) ⊂ Γ0 (M), the operators E R/A and E define
bi-distributions E R/A , E ∈ Γ0 (M 2 ) which we denote by the same symbol via e.g.
.
E R/A ( f, g) = f, E R/A g =
dg x dg y E R/A (x, y) f (x)g(y) .
M2
In terms of integral kernels of these distributions, some of the identities stated in
Theorem 2.1 read
Px E R/A (x, y) = δ(x, y) ,
E A (x, y) = E R (y, x) ,
E(y, x) = −E(x, y) .
The support properties of E R/A entail that E( f, g) vanishes if the supports of f
and g are spacelike separated. On the level of distribution kernels, this implies that
E(x, y) vanishes for spacelike separated x and y. In anticipation of the quantization
of the free Klein-Gordon field, this qualifies E(x, y) as a commutator function. In
the classical theory instead, E(x, y) defines a Poisson bracket or symplectic form.
To see this, we first need to specify the vector space on which this bracket should be
evaluated.
Definition 2.3 By Sol (Solsc ) we denote the space of real (spacelike-compact) solutions of the Klein-Gordon equation
.
Sol = {φ ∈ Γ (M) | Pφ = 0} ,
.
Solsc = Sol ∩ Γsc (M) .
By E we denote the quotient space
.
E = Γ0 (M)/P[Γ0 (M)] ,
which is the labelling space of linear on-shell observables of the free neutral KleinGordon field.
The fact that E is the labelling space of (classical) linear on-shell observables of the
free neutral Klein-Gordon field follows from the observation that each equivalence
class [ f ] ∈ E defines a linear functional on Sol by
.
Sol φ → O[ f ] (φ) = f, φ ,
where we note that, in the classical theory, Sol plays the role of the space of pure
states of the model. As φ is a solution of the Klein-Gordon equation O[ f ] (φ) does
not depend on the representative f ∈ [ f ] and is well-defined. The observable f, φ
may be interpreted as the ‘smeared classical field’ φ( f ) f, φ. The classical
observable φ(x), i.e. the observable that gives the value of a configuration φ at the
point x, may be obtained by formally considering φ( f ) with f = δx .
2.2 Linear Classical Fields on Curved Spacetimes
23
We know that every φ ∈ Sol is in one-to-one correspondence with initial data given
on an arbitrary but fixed Cauchy surface Σ of (M, g). Analogously the support of a
representative f ∈ [ f ] ∈ E can be chosen to lie in an arbitrarily small neighbourhood
of an arbitrary Cauchy surface.
Lemma 2.1 Let [ f ] ∈ E be arbitrary and let Σ be any Cauchy surface of (M, g).
Then, for any bounded neighbourhood O(Σ) of Σ, we can find a g ∈ Γ0 (M) with
supp g ⊂ O(Σ) and g ∈ [ f ].
Proof Let us assume that O(Σ) lies in the future of supp f , i.e. J − (supp f, M) ∩
O(Σ) = ∅, the other cases can be treated analogously. Let us consider two auxiliary
Cauchy surfaces Σ1 and Σ2 which are both contained in O(Σ) and which are chosen
such that Σ2 lies in the future of Σ whereas Σ1 lies in the past of Σ. Moreover, let us
take a smooth function χ ∈ Γ (M) which is identically vanishing in the future of Σ2
.
and fulfils χ ≡ 1 in the past of Σ1 and let us define g = f − Pχ E R f . By construction
and on account of the properties of both a globally hyperbolic spacetime (M, g) and a
retarded fundamental solution E R on M, χ E R f has compact support, hence g ∈ [ f ].
Finally, supp g is contained in a compact subset of J + (supp f, M) ∩ O(Σ).
We now observe that the causal propagator E induces a meaningful Poisson
bracket on E .
Proposition 2.1 The tuple (E , τ ) with τ : E × E → R defined by
.
τ ([ f ], [g]) = f, Eg
is a symplectic space. In particular
1. τ is well-defined and independent of the chosen representatives,
2. τ is antisymmetric,
3. τ is (weakly) non-degenerate, i.e. τ ([ f ], [g]) = 0 for all [g] ∈ E implies [ f ] =
[0].
Proof τ is independent of the chosen representatives because P ◦ E = 0. τ is
antisymmetric because E is formally skew-adjoint, cf. the last item of Theorem 2.1,
and because ·, · is symmetric. The non-degeneracy of τ follows again from the last
item of Theorem 2.1 and the fact that ·, · is non-degenerate.
In standard treatments on scalar field theory, one usually defines Poisson brackets
at ‘equal times’, but as realised by Peierls in [97], one can give a covariant version
of the Poisson bracket which does not depend on a splitting of spacetime into space
and time, and this is what we have given above. To relate the covariant form τ to an
equal-time version, we need the definition of a ‘future part’ of a function f ∈ Γ (M).
Definition 2.4 We consider a temporal cutoff function χ of the form discussed in
the proof of Lemma 2.1, i.e. a smooth function χ which is identically vanishing in
the future of some Cauchy surface Σ2 and identically one in the past of some Cauchy
24
2 Algebraic Quantum Field Theory on Curved Spacetimes
surface Σ1 in the past of Σ2 . Given such a χ , we define for an arbitrary f ∈ Γ (M)
the future part f + and the past part f − by
.
f + = (1 − χ ) f ,
f− = χf .
The relation of the covariant picture to the equal time-picture can be now shown
in several steps.
Theorem 2.2 Let ·, ·Sol be defined on tuples of solutions with compact overlapping
support by
. Sol × Sol (φ1 , φ2 ) → φ1 , φ2 Sol = Pφ1+ , φ2 .
Moreover, let Σ be an arbitrary Cauchy surface of (M, g) with future-pointing unit
normal vectorfield N and canonical measure dΣ induced by dg x .
1. The causal propagator E : Γ0 (M) → Γsc (M) descends to a bijective map
E : E → Solsc .
2. ·, ·Sol is antisymmetric and well-defined on all tuples of solutions with compact
overlapping support, in particular this bilinear form does not depend on the
choice of cutoff χ entering the definition of the future part.
3. For all f ∈ Γ0 (M) and all φ ∈ Sol, f, φ = E f, φSol . In particular, ·, ·Sol
is well-defined on all tuples of solutions with spacelike-compact overlapping
support.
4. For all f, g ∈ Γ0 (M), τ ([ f ], [g]) = E f, EgSol , thus the causal propagator
E : Γ0 (M) → Γsc (M) descends to an isomorphism between the symplectic
spaces (E , τ ) and (Solsc , ·, ·Sol ).
5. For all φ1 , φ2 ∈ Sol with spacelike-compact overlapping support,
φ1 , φ2 Sol =
dΣ N μ jμ (φ1 , φ2 ) ,
.
jμ (φ1 , φ2 ) = φ1 ∇μ φ2 − φ2 ∇μ φ1 .
Σ
6. For all f ∈ Γ0 (Σ) it holds
∇ N E f |Σ = f ,
E f |Σ = 0 .
On the level of distribution kernels, this entails that
∇ N E(x, y)|Σ×Σ = δΣ (x, y) ,
E(x, y)|Σ×Σ ≡ 0 ,
where δΣ is the δ-distribution with respect to the canonical measure on Σ.
Proof We sketch the proof. The first statement follows from the last item of
Theorem 2.1. The fact that ·, ·Sol is well-defined follows from the observation that
two different definitions φ + , φ + of the future part differ by a compactly supported
smooth function f = φ1+ − φ1+ ; consequently the supposedly different definitions
2.2 Linear Classical Fields on Curved Spacetimes
25
of the bilinear form differ by φ1 , φ2 Sol − φ1 , φ2 Sol = P f, φ2 = f, Pφ2 = 0.
Note that this partial integration is only possible because f has compact support, in
particular, ·, ·Sol is non-vanishing in general. The antisymmetry of ·, ·Sol follows
by similar arguments and Pφ + = P(φ −φ − ) = −Pφ − . The third statement follows
from the fact that E R f is a valid future part of E f , thus E f, φSol = P E R f, φ =
f, φ. The fourth statement follows immediately from the first and third one, whereas
the fifth one follows from ∇ μ jμ (φ1 , φ2 ) = φ2 Pφ1 − φ1 Pφ2 by an application of
Stokes theorem, see e.g. [38], where also a proof of the last statement can be found.
We now interpret the previous results. As argued above, elements [ f ] ∈ E label
linear on-shell observables φ( f ) f, φ, i.e. the classical field φ smeared with the
test function f . The causal propagator E induces a non-degenerate symplectic form τ
on E , which we may interpret as {φ( f ), φ(g)} τ ([ f ], [g]) = f, Eg, or, formally,
as {φ(x), φ(y)} = E(x, y). On the other hand, since (E , τ ) and (Solsc , ·, ·Sol ) are
symplectically isomorphic, we can equivalently label linear on-shell observables by
Solsc u, i.e. by u, φSol , the classical field ‘symplectically smeared’ with the
test solution u, where this symplectic smearing consists of integrating a particular
expression at equal times. The last result of the above theorem implies that the
covariant Poisson bracket {φ(x), φ(y)} = E(x, y) has the well-known equal-time
equivalent
{∇ N φ(x)|Σ , φ(y)|Σ } = ∇ N E(x, y)|Σ×Σ = δΣ (x, y) ,
{φ(x)|Σ , φ(y)|Σ } = E(x, y)|Σ×Σ = 0 ,
which may be interpreted as equal-time Poisson brackets of the field φ(x) and its
‘canonical momentum’ ∇ N φ(x). Further details on the relation between the equaltime and covariant picture can be found e.g. in [123, Chap. 3].
2.2.1.2 General Models Without Gauge Symmetry
The previous discussion of the classical free neutral Klein-Gordon field revealed
the essential ingredients defining this model. Following e.g. [6, 107], this can be
generalised to define an arbitrary linear field-theoretic model on a curved spacetime.
Definition 2.5 A real Bosonic linear field-theoretic model without local gauge symmetries on a curved spacetime is defined by the data (M, V , P), where
1. M (M, g) is a globally hyperbolic spacetime,
2. V is a real vector bundle over M, the space of smooth sections Γ (V ) of V
is endowed with a symmetric and non-degenerate bilinear form ·, ·V which
is well-defined on sections with compact overlapping support and given by the
integral of a fibrewise symmetric and non-degenerate bilinear form ·, ·V :
Γ (V ) × Γ (V ) → Γ (M),
3. P : Γ (V ) → Γ (V ) is a Green-hyperbolic partial differential operator, i.e. there
exist unique advanced E RP and retarded E AP fundamental solutions of P which
26
2 Algebraic Quantum Field Theory on Curved Spacetimes
P
P
P
satisfy P ◦ E R/A
= E R/A
◦ P = id|Γ0 (V ) and supp E R/A
f ⊂ J ± (supp f, M)
for all f ∈ Γ0 (V ); moreover P is formally self-adjoint with respect to ·, ·V .
A real Fermionic linear field-theoretic model without local gauge symmetries on a
curved spacetime is defined analogously with the only difference being that ·, ·V
and ·, ·V are not symmetric but antisymmetric. Complex theories can be obtained
from the real ones by complexification.
The relevance of the given data is as follows. Classical configurations Φ of the
linear field model under consideration are smooth sections Φ ∈ Γ (V ) of the vector
bundle V . We recall that V is locally of the form M × V with a real vector space V
which implies that locally Φ is a smooth function from M to V , see e.g. [82, 94] for
background material on vector bundles. We shall denote by Γ0 (V ), Γtc (V ), Γsc (V )
the subspaces of Γ0 (V ) consisting of smooth sections of V with compact, timelikecompact and spacelike-compact support, respectively.
The operator P specifies the equation of motion for the field model, the formal
self-adjointness of P is motivated by the fact that equations of motion arising as EulerLagrange equations of a Lagrangean are generally given by a formally self-adjoint
P. In fact the (formal) action S(Φ) = 21 Φ, PΦV leads to the Euler-Lagrange
equation PΦ = 0.
In the Klein-Gordon case we are dealing with an operator which is normally
hyperbolic, i.e. the leading order term is of the form −g μν ∂μ ∂ν . As reviewed in Theorem 2.1, this operator has a well-defined Cauchy problem, i.e. it is Cauchy-hyperbolic,
and consequently unique advance and retarded fundamental solutions exist such that
the operator is Green-hyperbolic. Example of partial differential operators which
are Cauchy-hyperbolic, but not normally hyperbolic are the Dirac operator and the
Proca operator which defines the equation of motion for a massive vector field, see
e.g. [6]. On the other hand, the distinction between Cauchy-hyperbolic operators
and Green-hyperbolic operators does not matter in most examples although one can
construction operators which are Green-hyperbolic but not Cauchy-hyperbolic, cf.
[6] for details.
Based on the data given in Definition 2.5, a symplectic space (Bosonic case) or
inner product space (Fermionic case) can be constructed in full analogy to the KleinGordon case, in particular, the following can be shown.
.
Theorem 2.3 Under the assumptions of Definition 2.5, let E P = E RP − E AP denote
the causal propagator of P, and let Sol ⊂ Γ (V ), Solsc ⊂ Γsc (V ) denote the space
of smooth (smooth and spacelike-compact) solutions of PΦ = 0 .
1. The tuple (E , τ ), where
.
E = Γ0 (V )/P[Γ0 (V )] ,
τ : E × E → R,
.
τ ([ f ], [g]) = f, E P g
V
,
is a well-defined symplectic (Bosonic case) or inner product (Fermionic case)
space. In particular τ is well-defined and independent of the chosen represen-
2.2 Linear Classical Fields on Curved Spacetimes
27
tatives and moreover non-degenerate and antisymmetric (Bosonic case) or symmetric (Fermionic case).
2. Let [ f ] ∈ E be arbitrary and let Σ be any Cauchy surface of (M, g). Then,
for any bounded neighbourhood O(Σ) of Σ, we can find a g ∈ Γ0 (V ) with
supp g ⊂ O(Σ) and g ∈ [ f ].
3. The causal propagator E P : Γ0 (V ) → Γsc (V ) descends to a bijective map
E → Solsc and for all f ∈ Γ0 (V ) and all Φ ∈ Sol,
f, ΦV = E P f, Φ
Sol
,
where for all Φ1 , Φ2 ∈ Sol with spacelike-compact overlapping support, the
bilinear form ·, ·Sol is defined as
. Φ1 , Φ2 Sol = PΦ1+ , Φ2 V .
4. ·, ·Sol may be computed as a suitable integral over an arbitrary but fixed Cauchy
surface Σ of (M, g) with future-pointing normal vector field N and induced
measure dΣ. If there exists a ‘current’ j : Γ (V ) × Γ (V ) → T ∗ M such that
∇ μ jμ (Φ1 , Φ2 ) = Φ1 , PΦ2 V − Φ2 , PΦ1 V for all Φ1 , Φ2 ∈ Γ (V ), then
Φ1 , Φ2 Sol =
dΣ N μ jμ (Φ1 , Φ2 ) .
Σ
5. The tuple (Solsc , ·, ·Sol ) is a well-defined symplectic (Bosonic case) or inner
product (Fermionic case) space which is isomorphic to (E , τ ).
As with the Klein-Gordon field, the last statement implies in physical terms that
the symplectic respectively inner product space can be constructed both in a covariant
and in an equal-time fashion, and that the two constructions give equivalent results.
In many cases, the equal-time point of view is better suited for practical computations
and for proving particular further properties of the bilinear form τ , cf. the following
discussion of theories with local gauge invariance.
2.2.2 Models with Gauge Symmetry
The discussion of linear field theoretic models with local gauge symmetries on curved
spacetimes is naturally more involved than the case where such symmetries are
absent. However, as in this monograph we will only be dealing with linear models
and simple observables, it will not be necessary to introduce auxiliary fields like in
the BRST/BV formalism [50, 68]. Instead, we shall review an approach which has
been developed in [39] for the Maxwell field, used for linearised gravity in [42] and
then further generalised to arbitrary linear gauge theories in [63]. For linear models
and simple observables this approach and the BRST/BV formalism give equivalent
28
2 Algebraic Quantum Field Theory on Curved Spacetimes
results, however, non-linear models and more general observables are not tractable
in the way we shall review in the following.
2.2.2.1 A Toy Model
We outline the essential ideas of this approach at the example of a toy model. We
consider as a gauge field Φ = (φ1 , φ2 )t ⊂ Γ (V ) a tuple of two scalar fields on a
spacetime (M, g) satisfying the equation of motion
PΦ =
− φ1
= 0,
φ2
−
.
where V is the (trivial) vector bundle V = M × R2 . The gauge transformations are
given by the following translations on configuration space Φ → Φ + K ε, where the
gauge transformation operator K : Γ (M) → Γ (V ) is the linear operator defined
.
by K ε = (ε, ε)t for a smooth function ε ∈ Γ (M). One may check that P ◦ K = 0
.
holds which is equivalent
to the gauge-invariance of the action S(Φ) = 21 Φ, PΦV
. with Φ, Φ V = M dg x (φ1 φ1 + φ2 φ2 ).
.
Clearly, the linear combination ψ = φ1 − φ2 is gauge-invariant and satisfies
−ψ = 0, and it would be rather natural to quantize Φ by directly quantizing
ψ as a massless, minimally coupled scalar field. This would be much in the spirit
of the usual quantization of perturbations in Inflation, where gauge-invariant linear combinations of the gauge field components, e.g. the Bardeen-Potentials or the
Mukhanov-Sasaki variable, are taken as the fundamental fields for quantization,
see the last chapter of this monograph. However, in general it is rather difficult
to directly identify a gauge-invariant fundamental field like ψ whose classical and
quantum theory is equivalent to the classical and quantum theory of the original
gauge field. Notwithstanding, an indirect characterisation of such a gauge-invariant
linear combination of gauge-field components, which can serve as a fundamental
field for quantization, is still possible. In the toy model under consideration we consider a tuple f = ( f 1 , f 2 ) ∈ Γ0 (V ) of test functions f i ∈ Γ0 (M). We ask that
.
†
K † f = f 1 + f 2 = 0, where K
: Γ (V )† → Γ (M) is the adjoint of the gauge
transformation operator K i.e. M dg x εK f = K ε, f V . Clearly, any f satisfying this condition is of the form f = (h, −h)t for a test function h. We now
observe that the pairing between a gauge field configuration
Φ and such an f is
gauge-invariant, i.e. Φ + K ε, f V = f, ΦV + M dg x εK † f = Φ, f V . Thus
we can consider the ‘smeared field’ Φ( f ) f, ΦV , with f = (h, −h)t and arbitrary h, as a gauge-invariant linear combination of gauge-field components which is
suitable for playing
the role of a fundamental field for quantization. We can compute f, ΦV = M dg x ψh, and observe that, up to the ‘smearing’ with h, this
indirect choice of gauge-invariant fundamental field is exactly the one discussed in
the beginning. If one chooses h to be the delta distribution δ(x, y) rather than a
test function, one even finds f, Φ = ψ(x), whereas for general h, f, Φ can be
interpreted as a weighted, gauge-invariant measurement of the field configuration Φ.
2.2 Linear Classical Fields on Curved Spacetimes
29
Moreover, as already anticipated, in general gauge theories with more complicated
gauge transformation operators K it usually extremely difficult to classify all solutions of K † f = 0, which would be equivalent to a direct characterisation of one or
several fundamental gauge-invariant fields such as ψ, whereas working implicitly
with the condition K † f = 0 is always possible.
2.2.2.2 General Models
From the previous discussion we can already infer most of the additional data which
is needed in addition to the data mentioned in Definition 2.5 in order to specify a
linear field theoretic model with local gauge symmetries on curved spacetimes.
Definition 2.6 A real Bosonic linear field-theoretic model with local gauge symmetries on a curved spacetime is defined by the data (M, V , W , P, K ), where
1. M (M, g) is a globally hyperbolic spacetime,
2. V and W are real vector bundles over M, the spaces of smooth sections V and
W are endowed with symmetric and non-degenerate bilinear forms ·, ·V and
·, ·W which are well-defined on sections with compact overlapping support and
given by the integral of fibrewise symmetric and non-degenerate bilinear forms
·, ·V : Γ (V ) × Γ (V ) → Γ (M) and ·, ·W : Γ (W ) × Γ (W ) → Γ (M),
3. P : Γ (V ) → Γ (V ) is a partial differential operator which is formally selfadjoint with respect to ·, ·V ,
4. K : Γ (W ) → Γ (V ) is a partial differential operator such that P ◦ K = 0;
.
moreover R = K † ◦ K : Γ (W ) → Γ (W ) is Cauchy-hyperbolic and there exists
.
=
an operator T : Γ (W ) → Γ (V ) such that a) P
P +T ◦K † is Green-hyperbolic
. †
and b) Q = K ◦ T is Cauchy-hyperbolic.
A real Fermionic linear field-theoretic model with local gauge symmetries on a curved
spacetime is defined analogously with the only difference being that ·, ·V , ·, ·V ,
·, ·W and ·, ·W are not symmetric but antisymmetric. Complex theories can be
obtained from the real ones by complexification.
These data have the following meaning. Sections of V are configurations of the
gauge field Φ, whereas local gauge transformations are parametrised via the gauge
transformation operator K by sections of W . The differential operator P defines the
equation of motion for the gauge field Φ via PΦ = 0. The formal self-adjointness of
P is motivated by P being the Euler-Lagrange operator of a local action S(Φ), e.g.
S(Φ) = 21 Φ, PΦV , whereas the gauge-invariance condition P ◦ K = 0 implies
gauge-invariance of the action S(Φ). This condition implies (for K = 0) that P
can not be Cauchy-hyperbolic, because any ‘pure gauge configuration’ Φε = K ε
with ε ∈ Γ0 (W ) of compact support solves the equation of motion PΦε = 0 with
vanishing initial data in the distant past, whereas for Cauchy-hyperbolic P the unique
solution with vanishing initial data is identically zero.
The Cauchy-hyperbolicity of R = K † ◦ K implies that for every Φ ∈ Γ (V ) there
.
exists an ε ∈ Γ (W ) such that Φ = Φ + K ε satisfies the ‘canonical gauge-fixing
30
2 Algebraic Quantum Field Theory on Curved Spacetimes
condition’ K † Φ = 0. The existence of the gauge-fixing operator T such that the
= P + T ◦ K † is Green-hyperbolic
gauge-fixed equation of motion operator P
= 0 up to gaugeimplies that every solution of PΦ = 0 in fact satisfies PΦ
equivalence; consequently, the dynamics of the ‘physical degrees of freedom’ is ruled
by a hyperbolic equation of motion even if P is not hyperbolic. Finally, the condition
that Q = K † ◦ T is Cauchy-hyperbolic implies that the gauge-fixing K † Φ = 0 is
= 0. T is in general not canonical
compatible with the hyperbolic dynamics of PΦ
and the following constructions will not depend on the particular choice of T in case
several T with the required properties exist, thus we do not consider the gauge-fixing
operator T as part of the data specifying the model.
Apart from the toy model discussed above, a simple example of a linear gauge
theory which fits into Definition 2.6 is the Maxwell field (on a trivial principal U (1)bundle) which after all was the inspiration for the formulation of this definition. This
model is specified by (in differential form notation)
W = M × R,
V = W ⊗ T∗M = T ∗M ,
.
.
ε1 , ε2 W = ε1 ∧ ∗ε2 ,
Φ1 , Φ2 V = Φ1 ∧ ∗Φ2 ,
M
M
P = d †d ,
= d † d + dd † ,
P
K =T =d,
K † K = K † T = d †d = .
We would like to construct a (pre-)symplectic or (pre-)inner product space corresponding to the data given in Definition 2.6 by following as much as possible the logic
of the case without gauge symmetry. To this avail we need a few further definitions
of section spaces.
Definition 2.7 As before, we denote by Sol ⊂ Γ (V ) (Solsc ⊂ Γsc (V )) the spaces
of smooth solutions of the equation PΦ = 0 (with spacelike-compact support). By
G and Gsc we denote the space of gauge configurations (with spacelike-compact
support), by Gsc,0 we denote the gauge configurations induced by spacelike-compact
gauge transformation parameters
.
G = K [Γ (W )] ,
.
Gsc = G ∩ Γsc (W ) ,
.
Gsc,0 = K [Γsc (W )] .
In general, Gsc,0 Gsc . By ker 0 (K † ) we denote the space of gauge-invariant testsections and by E the labelling space of linear gauge-invariant on-shell observables
.
ker 0 (K † ) = { f ∈ Γ0 (V ) | K † f = 0} ,
.
E = ker 0 (K † )/P [Γ0 (V )] .
Our discussion of the toy model in the previous subsection already indicated
why E defined above is a good candidate for a labelling space of linear gaugeinvariant on-shell observables. First of all we observe that E is well-defined because
2.2 Linear Classical Fields on Curved Spacetimes
31
P [Γ0 (V )] ⊂ ker 0 (K † ) owing to P ◦ K = 0. Moreover, we have by construction
for arbitrary Φ ∈ Sol, ε ∈ Γ (W ), f ∈ ker 0 (K † ) and g ∈ Γ0 (V )
f + Pg, Φ + K εV = f, ΦV .
Consequently, every element [ f ] of E induces a well-defined linear functional on
Sol/G , i.e. on gauge-equivalence classes of on-shell configurations, by Sol/G .
[Φ] → O[ f ] ([Φ]) = f, ΦV . Being gauge-invariant, these functionals correspond
to meaningful (physical) observables. On the level of classical observables, the fact
that the physical degrees of freedom of the gauge field propagate in a causal fashion
is reflected in the following generalisation of Lemma 2.1 which is proved in [63].
Lemma 2.2 Let [ f ] ∈ E be arbitrary and let Σ be any Cauchy surface of (M, g).
Then, for any bounded neighbourhood O(Σ) of Σ, we can find a g ∈ ker 0 (K † ) with
supp g ⊂ O(Σ) and g ∈ [ f ].
In constructing the classical bracket for models without gauge symmetry the last
statement of Theorem 2.1, which in fact holds for the causal propagator E P of any
Green-hyperbolic operator P on an arbitrary vector bundle V , has been crucial. In
the following, we review results obtained in [63, Theorem 3.12+Theorem 5.2] which
essentially imply that, although P is not hyperbolic, the causal propagator E P of the
= P + T ◦ K † is effectively a causal
gauge-fixed equation of motion operator P
propagator for P up to gauge-equivalence. The crucial observation here is that P and
coincide on ker 0 (K † ) which implies that E P
restricted to ker 0 (K † ) is independent
P
of the particular form of the gauge fixing operator T .
= P + T ◦ K † satisfies the following
Theorem 2.4 The causal propagator E P of P
relations.
1. h ∈ ker 0 (K † ) and E P h ∈ Gsc,0 if and only if h ∈ P[Γ0 (V )], with Gsc,0 defined
in Definition 2.7.
2. Every h ∈ Solsc can be split as h = h 1 + h 2 with h 1 ∈ E P ker 0 (K † ) and
h 2 ∈ Gsc,0 .
3. E P descends to a bijective map E → Solsc /Gsc,0 .
4. E P is formally skew-adjoint w.r.t. ·, ·V on ker 0 (K † ), i.e.
h1, E P h2
V
= − E P h1, h2
V
for all h 1 , h 2 ∈ ker 0 (K † ).
5. Let T : Γ (W ) → Γ (V ) be any differential operator satisfying the properties
required for the operator T in Definition 2.6 and let E P be the causal propagator
.
= P + T ◦ K † . Then E P satisfies the four properties above.
of P
Given these results, we can now construct a meaningful bracket on E by generalising Proposition 2.1.
32
2 Algebraic Quantum Field Theory on Curved Spacetimes
Proposition 2.2 The tuple (E , τ ) with τ : E × E → R defined by
.
τ ([ f ], [g]) = f, E P g
V
is a pre-symplectic space (Bosonic case) or pre-inner product space (Fermionic case).
In particular,
1. τ is well-defined and independent of the chosen representatives,
2. τ is antisymmetric (Bosonic case) or symmetric (Fermionic case).
3. Let T : Γ (W ) → Γ (V ) be any differential operator satisfying the properties
required for the operator T in Definition 2.6 and define τ in analogy to τ but with
.
=
P + T ◦ K † instead of E P . Then τ = τ .
the causal propagator E P of P
We stress that τ is in general not weakly non-degenerate, cf. the last statement
of Theorem 2.5. This is a particular feature of gauge theories, cf. e.g. [8, 110] for
a discussion of the physical interpretation of this non-degeneracy in the case of the
Maxwell field. The last statement above indicates that τ is independent of the gaugefixing operator T and in this sense, gauge-invariant. Indeed, we shall see in what
follows that τ can be rewritten in a manifestly gauge-invariant form. The form of τ
given here can be derived directly from the action S(Φ) = 21 Φ, PΦV by Peierls’
method in analogy to the derivation for electromagnetism in [110], see also [79, 80])
for a broader context.
As in the case without local gauge symmetries it is interesting and useful to
observe that the covariant pre-symplectic or pre-inner product space (E , τ ) can be
understood equivalently in an equal-time fashion. In fact the following statements
have been proved in [63, Proposition 5.1+Theorem 5.2] (or can be proved by slightly
generalising the arguments used there).
Theorem 2.5 Under the assumptions of Definition 2.6, let ·, ·Sol be the bilinear
form on Sol defined for Φ1 , Φ2 ∈ Sol with spacelike-compact overlapping support by
. Φ1 , Φ2 Sol = PΦ1+ , Φ2 V ,
where Φ + denotes the future part of Φ, see Definition 2.4. This bilinear form has the
following properties.
1. Φ1 , Φ2 Sol is well-defined for all Φ1 , Φ2 ∈ Sol with spacelike-compact overlapping support. In particular, it is independent of the choice of future part entering
its definition.
2. ·, ·Sol is antisymmetric (Bosonic case) or symmetric (Fermionic case).
3. ·, ·Sol is gauge-invariant, i.e.
Φ1 , Φ2 + K εSol = Φ1 , Φ2 Sol
for all Φ1 , Φ2 ∈ Sol, ε ∈ Γ (W ) s.t. Φ1 and ε have spacelike-compact overlapping support.
2.2 Linear Classical Fields on Curved Spacetimes
33
4. ·, ·Sol may be computed as a suitable integral over an arbitrary but fixed Cauchy
surface Σ of (M, g) with future-pointing normal vector field N and induced
measure dΣ. If there exists a ‘current’ j : Γ (V ) × Γ (V ) → T ∗ M such that
∇ μ jμ (Φ1 , Φ2 ) = Φ1 , PΦ2 V − Φ2 , PΦ1 V for all Φ1 , Φ2 ∈ Γ (V ), then
Φ1 , Φ2 Sol =
dΣ N μ jμ (Φ1 , Φ2 ) .
Σ
5. For all Φ ∈ Sol and all h ∈ ker 0 (K † ),
E P h, Φ
Sol
= h, ΦV .
6. E P descends to an isomorphism of pre-symplectic (Bosonic case) or pre-inner
product spaces (Fermionic case) E P : (E , τ ) → (Solsc /Gsc,0 , ·, ·Sol ).
7. If Gsc,0 Gsc , then τ is degenerate, i.e. there exists [0] = [h] ∈ E s.t. τ ([h], E ) =
0.
The fourth and fifth statement in the above theorem show that one can view
the observable Sol/G [Φ] → h, ΦV Φ(h), i.e. the ‘covariantly smeared
classical field’, equivalently as an ‘equal-time smeared classical field’ Sol/G [Φ] → H, ΦSol with H = E P h ∈ [H ] ∈ Solsc /Gsc,0 .
2.3 Linear Quantum Fields on Curved Spacetimes
Given the (pre-)symplectic or (pre-)inner product spaces of classical linear observables constructed in the previous section for Bosonic and Fermionic theories with
or without local gauge symmetries, there are several ‘canonical’ ways to construct
corresponding algebras of observables in the associated quantum theories; these
constructions differ mainly in technical terms.
In the Bosonic case, one can consider the Weyl algebra corresponding to the
pre-symplectic space (E , τ ), which essentially means to quantize exponentials
exp(i f, ΦV ) of the smeared classical field Φ( f ) f, ΦV rather than the
smeared classical field itself. This mainly has the technical advantage that one is
dealing with a C ∗ -algebra corresponding to bounded operators on a Hilbert space,
i.e. to operators with a bounded spectrum. However, sometimes it is also advisable in
physical terms to consider exponential observables rather than linear ones as fundamental building blocks, e.g. in case one is dealing with finite gauge transformations,
cf. [8]. The Weyl algebra is initially constructed under the assumption that the form
τ is non-degenerate such that the pre-symplectic space (E , τ ) is in fact symplectic.
However, the construction of the Weyl algebra is also well-defined in the degenerate
case [12] and even in case E is not a vector space, but only an Abelian group [8]. In
the Fermionic case it is not necessary to consider exponential observables in order
34
2 Algebraic Quantum Field Theory on Curved Spacetimes
to access the advantages of a C ∗ -algebraic framework, as a C ∗ -algebra can already
be constructed based on linear observables due to the anticommutation relations of
Fermionic quantum fields, see e.g. [14].
As in this monograph we shall not make use of the C ∗ -algebraic framework, it
will be sufficient to consider the algebra of quantum observables constructed by
directly quantizing the smeared classical fields Φ( f ) f, ΦV themselves both
in the Bosonic case and in the Fermionic case. We shall first construct the BorchersUhlmann algebra A (M) corresponding to the linear model defined by the data
(M, V , W , P, K ) cf. Definition 2.6, where we can consider a model (M, V , P)
without gauge invariance as the subclass (M, V , W = V , P, K = 0). The algebra
A (M) contains only the most simple observables, in particular it does not contain
observables which correspond to pointwise powers of the field such as Φ(x)2 or
the stress-energy tensor. We shall discuss a larger algebra which contains also these
observables in Sect. 2.6.1.
In order to construct the Borchers-Uhlmann algebra, we recall that the labelling
space E consists of equivalence classes of test sections [ f ] corresponding to the
classical observables Φ( f ) f, ΦV with Φ ∈ [Φ] ∈ Sol/G (= Sol in the absence
of gauge symmetries, i.e. for K = 0) and f ∈ [ f ] ∈ E = ker 0 (K † )/P[Γ0 (V )]
(= Γ0 (V )/P[Γ0 (V )] for K = 0). We recall that omitting the equivalence classes
in the notation Φ( f ) f, ΦV is meaningful because the latter expression is
independent of the chosen representatives of the equivalence classes. For the quantum
theory, we need complex expressions and therefore consider the complexification
.
EC = E ⊗R C of the labelling space E .
With this in mind, we represent the quantum product of two smeared fields
Φ( f 1 )Φ( f 2 ) by the tensor product [ f 1 ] ⊗ [ f 2 ]. On the long run, we would like
to represent Φ( f ) as an operator on a Hilbert space, we therefore need an operation
which encodes ‘taking the adjoint with respect to a Hilbert space inner product’ on the
.
abstract algebraic level. We define such a ∗-operation by setting [Φ( f )]∗ = Φ( f ),
.
∗
∗
∗
corresponding to [ f 1 ] = [ f 1 ], and [Φ( f 1 ) · · · Φ( f n )] = [Φ( f n )] · · · [Φ( f 1 )]∗ .
Observables would then be polynomials P of smeared fields (tensor polynomials
of elements of EC ) which fulfil P ∗ = P (in the Fermionic case we need further
conditions e.g. P has to be an even polynomial). To promote Φ( f ) to a proper
quantum field with the correct (anti)commutation relations, we define
.
[Φ( f ), Φ(g)]∓ = Φ( f )Φ(g) ∓ Φ(g)Φ( f ) = iτ ([ f ], [g])I = i f, E P , g
V
I , (2.6)
where we recall that E P is the causal propagator of the gauge-fixed equation of
= P + T ◦ K † (= P in the absence of gauge symmetries K = 0),
motion operator P
I is the identity and the − (+) sign applies for the Bosonic (Fermionic) case. Recall
that τ ([ f ], [g]) vanishes if the supports of f and g are spacelike separated, the above
canonical (anti)commutation relations (CCR/CAR) therefore assure that observables
commute at spacelike separations. In the case without gauge symmetries, one can
write the CCR/CAR formally as
[Φ(x), Φ(y)]∓ = i E P (x, y)I .
2.3 Linear Quantum Fields on Curved Spacetimes
35
Recall that this is nothing but the covariant version of the well-known equal-time
CCR/CAR. In the case where gauge symmetries are present, this expression does not
make sense because the equality holds only when smeared with ‘gauge-invariant’
test sections f ∈ ker 0 (K † ). Finally, we remark that dynamics is already encoded
by the fact that EC consists of equivalence classes with [Pg] = [0] ∈ EC for all
g ∈ Γ0C (V ) and that it is convenient to have a topology on the algebra A (M) in
order to be able to quantify to which extent two abstract observables are ‘close’,
i.e. similar in physical terms. We subsume the above discussion in the following
definition.
Definition 2.8 Consider a linear Bosonic or Fermionic (gauge) field theory defined
by (M, V , W , P, K ) (with W = V and K = 0 in the absence of gauge symmetries),
cf. Definitions 2.5 and 2.6 and let (E , τ ) be the corresponding (pre-)symplectic or
(pre-)inner product space constructed as in Theorem 2.3 and Proposition 2.2. The
Borchers-Uhlmann algebra A (M) of the model (M, V , W , P, K ) is defined as
.
A (M) = A0 (M)/I ,
where A0 (M) is the direct sum
∞
. ⊗n
EC
A0 (M) =
n=0
.
(EC⊗0 = C) equipped with a product defined by the linear extension of the tensor
product of EC⊗n , a ∗-operation defined by the antilinear extension of ([ f 1 ] ⊗ · · · ⊗
[ f n ])∗ = [ f n ] ⊗ · · · ⊗ [ f 1 ], and it is required each element of A0 (M) is a linear
combination of elements of EC⊗n with n ≤ n max < ∞. Additionally, we equip A0 (M)
with the topology induces by the locally convex topology of Γ0 (V ). Moreover, I is
the closed ∗-ideal generated by elements of the form −iτ ([ f ], [g]) ⊕ ([ f ] ⊗ [g] ∓
[g] ⊗ [ f ]), where − (+) stands for the Bosonic (Fermionic) case, and A (M) is
thought to be equipped with the product, ∗-operation, and topology descending from
A0 (M). If O is an open subset of M, A (O) denotes the algebra obtained by allowing
only test sections with support in O.
A (M), in contrast to A0 (M), depends explicitly on the metric g of a spacetime
(M, g) via the causal propagator and the equation of motion. However, now and in
the following we shall omit this dependence in favour of notational simplicity.
We recall that, by Lemma 2.2, every equivalence class [ f ] ∈ EC is so large
that it contains elements with support in an arbitrarily small neighbourhood of any
Cauchy surface of M. This implies the following well-known result, which on physical grounds entails the predictability of observables.
Lemma 2.3 The Borchers-Uhlmann algebra A (M) fulfils the time-slice axiom.
Namely, let Σ be a Cauchy surface of (M, g) and let O be an arbitrary neighbourhood
of Σ. Then A (O) = A (M).
36
2 Algebraic Quantum Field Theory on Curved Spacetimes
We now turn our attention to states. Let A be a topological, unital ∗-algebra, i.e.
A is endowed with an operation ∗ which fulfils (AB)∗ = B ∗ A∗ and (A∗ )∗ = A for
all elements A, B in A. A state ω on A is defined to be a continuous linear functional
A → C which is normalised, i.e. ω(I) = 1 and positive, namely, ω(A∗ A) ≥ 0 must
hold for any A ∈ A. Considering the special topological and unital ∗-algebra A (M),
a state on A (M) is determined by its n-point correlation functions
.
ωn ( f 1 , . . . , f n ) = ω (Φ( f 1 ) · · · Φ( f n )) .
which are distributions in Γ0C (M n ). Given a state ω on A, one can represent A on a
Hilbert space Hω by the so-called GNS construction (after Gel’fand, Naimark, and
Segal), see for instance [4, 65]. By this construction, algebra elements A ∈ A are
represented as operators πω (A) on a common dense and invariant subspace of Hω ,
while ω is represented as a vector of |Ωω ∈ Hω such that for all A ∈ A
ω(A) = Ωω |πΩ (A)|Ωω .
Conversely, every vector in a Hilbert space H gives rise to an algebraic state on the
algebra of linear operators on H .
Among the possible states on A (M) there are several special classes, which we
collect in the following definition. Some of the definitions are sensible for general
∗-algebras, as we point out explicitly.
Definition 2.9 Let A denote a general ∗-algebra and let A (M) denote the BorchersUhlmann algebra of a linear Bosonic or Fermionic (gauge) field theory.
1. A state ω on A is called mixed, if it is a convex linear combination of states, i.e.
ω = λω1 + (1 − λ)ω2 , where λ < 1 and ωi = ω are states on A. A state is called
pure if it is not mixed.
2. A state ω on A (M) is called even, if it is invariant under Φ( f ) → −Φ( f ), i.e.
it has vanishing n-point functions for all odd n.
3. An even state on A (M) is called quasifree or Gaussian if, for all even n,
ωn ( f 1 , . . . , f n ) =
n/2
ω2 f πn (2i−1) , f πn (2i) .
πn ∈Sn i=1
Here, Sn denotes the set of ordered permutations of n elements, namely, the
following two conditions are satisfied for πn ∈ Sn :
πn (2i − 1) < πn (2i) for 1 ≤ i ≤ n/2 ,
πn (2i − 1) < πn (2i + 1) for 1 ≤ i < n/2 .
4. Let αt denote a one-parameter group of ∗-automorphisms on A, i.e. for arbitrary
elements A, B of A,
2.3 Linear Quantum Fields on Curved Spacetimes
αt A∗ B = (αt (A))∗ αt (B) ,
37
αt (αs (A)) = αt+s (A) ,
α0 (A) = A .
A state ω on A is called αt -invariant if ω(αt (A)) = ω(A) for all A ∈ A.
5. An αt -invariant state ω on A is said to satisfy the KMS condition for an inverse
temperature β = T −1 > 0 if, for arbitrary elements A, B of A, the two functions
.
FAB (t) = ω (Bαt (A)) ,
.
G AB (t) = ω (αt (A)B)
extend to functions FAB (z) and G AB (z) on the complex plane which are analytic
in the strips 0 < Im z < β and −β < Im z < 0 respectively, continuous on the
boundaries Im z ∈ {0, β}, and fulfil
FAB (t + iβ) = G AB (t) .
The KMS condition (after Kubo, Martin, and Schwinger) holds naturally for Gibbs
states of finite systems in quantum statistical mechanics, i.e. for states that are given
as ωβ (A) = Trρ A with a density matrix ρ = exp (−β H )(Tr exp (−β H ))−1 , H the
Hamiltonian operator of the system, and Tr denoting the trace over the respective
Hilbert space. This follows by setting
αt (A) = eit H Ae−it H ,
making use of the cyclicity of the trace, and considering that exp (−β H ) is bounded
and has finite trace in the case of a finite system. In the thermodynamic limit,
exp (−β H ) does not possess these properties any more, but the authors of [64]
have shown that the KMS condition is still a reasonable condition in this infinitevolume limit. Physically, KMS states are states which are in (thermal) equilibrium
with respect to the time evolution encoded in the automorphism αt . In general curved
spacetimes, there is no ‘time evolution’ which acts as an automorphism on A (M).
One could be tempted to introduce a time evolution by a canonical time-translation
with respect to some time function of a globally hyperbolic spacetime. However, the
causal propagator E P will in general not be invariant under this time translation if
the latter does not correspond to an isometry of (M, g). Hence, such time-translation
would not result in an automorphism of A (M). There have been various proposals
to overcome this problem and to define generalised notions of thermal equilibrium
in curved spacetimes, see [119] for a review. Ground states (vacuum states) may be
thought of as KMS states with inverse temperature β = T −1 = ∞.
Quasifree or Gaussian states which are in addition pure are closely related to the
well-known Fock space picture in the sense that the GNS representation of a pure
quasifree state is an irreducible representation on Fock space, see e.g. [78, Sect. 3.2].
In this sense, a pure quasifree state is in one-to-one correspondence to a specific
definition of a ‘particle’.
For the remainder of this chapter and most of the remainder of this monograph,
the only model we shall discuss is the free neutral Klein-Gordon field for simplicity.
However, all concepts we shall review will be applicable to more general models.
38
2 Algebraic Quantum Field Theory on Curved Spacetimes
2.4 Hadamard States
The power of the algebraic approach lies in its ability to separate the algebraic
relations of quantum fields from the Hilbert space representations of these relations
and thus in some sense to treat all possible Hilbert space representations at once.
However, the definition of an algebraic state reviewed in the previous subsection is
too general and thus further conditions are necessary in order to select the physically
meaningful states among all possible ones on A (M).
To this avail it seems reasonable to look at the situation in Minkowski spacetime.
Physically interesting states there include the Fock vacuum state and associated multiparticle states as well as coherent states and states describing thermal equilibrium
situations. All these states share the same ultraviolet (UV) properties, i.e. the same
high-energy behaviour, namely they satisfy the so-called Hadamard condition, which
we shall review in a few moments. A closer look at the formulation of quantum field
theory in Minkowski spacetime reveals that the Hadamard condition is indeed essential for the mathematical consistency of QFT in Minkowski spacetime, as we would
like to briefly explain now. In the following we will only discuss real scalar fields for
simplicity. Analyses of Hadamard states for fields of higher spin can be found e.g.
in [34, 43, 67, 107].
The Borchers-Uhlmann algebra A (M) of the free neutral Klein-Gordon field φ
contains only very basic observables, namely, linear combinations of products of free
fields at separate points, e.g. φ(x)φ(y). However, if one wants to treat interacting
fields in perturbation theory, or the backreaction of quantum fields on curved spacetimes via their stress-energy tensor, ones needs a notion of normal ordering, i.e. a
way to define field monomials like φ 2 (x) at the same point. To see that this requires
some work, let us consider the massless scalar field in Minkowski spacetime. Its two
point function reads
ω2 (x, y) = ω(φ(x)φ(y)) = lim
ε↓0
1
1
,
2
2
4π (x − y) + 2 iε(x0 − y0 ) + ε2
(2.7)
where (x − y)2 denotes the Minkowskian product induced by the Minkowski metric
and the limit has to be understood as being performed after integrating ω2 with at
least one test function (weak limit). ω2 (x, y) is a smooth function if x and y are
spacelike or timelike separated. It is singular at (x − y)2 = 0, but the singularity is
‘good enough’ to give a finite result when smearing ω2 (x, y) with two test functions.
Hence, ω2 is a well-defined (tempered) distribution. Loosely speaking, this shows
once more that the product of fields φ(x)φ(y) is ‘well-defined’ at non-null related
points. However, if we were to define φ 2 (x) by some ‘limit’ like
.
φ 2 (x) = lim φ(x)φ(y) ,
x→y
2.4 Hadamard States
39
the expectation value of the resulting object would ‘blow up’ and would not be any
meaningful object. The well-known solution to this apparent problem is to define
field monomials by appropriate regularising subtractions. For the squared field, this
is achieved by setting
.
: φ 2 (x) : = lim (φ(x)φ(y) − ω2 (x, y)I) ,
x→y
where of course one would have to specify in which sense the limit should be taken.
Omitting the details of this procedure, it seems still clear that the Wick square : φ 2 (x) :
is a meaningful object, as it has a sensible expectation value, i.e. ω(: φ 2 (x) :) = 0. In
the standard Fock space picture, one heuristically writes the field (operator) in terms
of creation and annihilation operators in momentum space, i.e.
1
φ(x) = √ 3
2π
R3
d k † ikx
a e + a e−ikx ,
√
k
k
2k0
and defines : φ 2 (x) : by writing the mode expansion of the product φ(x)φ(y), ‘normal
ordering’ the appearing products of creation and annihilation operators such that the
creation operators are standing on the left hand side of the annihilation operators, and
then finally taking the limit x → y. It is easy to see that this procedure is equivalent
to the above defined subtraction of the vacuum expectation value. However, having
defined the Wick polynomials is not enough. We would also like to multiply them,
i.e., we would like them to constitute an algebra. Using the mode-expansion picture,
one can straightforwardly compute
: φ 2 (x) :: φ 2 (y) : = : φ 2 (x)φ 2 (y) : +4 : φ(x)φ(y) : ω2 (x, y) + 2 (ω2 (x, y))2 ,
which is a special case of the well-known Wick theorem, see for instance [76]. The
right hand side of the above equation is a sensible object if the appearing square
of the two-point function ω2 (x, y) is well-defined. In more detail, we know that
ω2 (x, y) has singularities, and that these are integrable with test functions. Obviously, (ω2 (x, y))2 has singularities as well, and the question is whether the singularities are still good enough to be integrable with test functions. In terms of a mode
decomposition, one could equivalently wonder whether the momentum space integrals appearing in the definition of : φ 2 (x)φ 2 (y) : via normal ordering creation and
annihilation operators converge in a sensible way. The answer to these questions is
‘yes’ because of the energy positivity property of the Minkowskian vacuum state,
and this is the reason why one usually never worries about whether normal ordering
is well-defined in quantum field theory on Minkowski spacetime. In more detail, the
Fourier decomposition of the massless two-point function ω2 reads
1
ω2 (x, y) = lim
ε↓0 (2π )3
R4
dk Θ(k0 )δ(k 2 ) eik(x−y) e−εk0 ,
(2.8)
40
2 Algebraic Quantum Field Theory on Curved Spacetimes
where Θ(k0 ) denotes the Heaviside step function. We see that the Fourier transform of
ω2 has only support on the forward lightcone (or the positive mass shell in the massive
case); this corresponds to the fact that we have associated the positive frequency
modes to the creation operator in the above mode expansion of the quantum field. This
insight allows to determine (or rather, define) the square of ω2 (x, y) by a convolution
in Fourier space
(ω2 (x, y))2 = lim
ε↓0
1
(2π)6
1
= lim
ε↓0 (2π)6
dq
R4
R4
dk
R4
dp Θ(q0 ) δ(q 2 ) Θ( p0 ) δ( p2 ) ei(q+ p)(x−y) e−ε(q0 + p0 )
dq Θ(q0 ) δ(q 2 ) Θ(k0 − q0 ) δ((k − q)2 ) eik(x−y) e−εk0 .
R4
Without going too much into details here, let us observe that the above expression
can only give a sensible result (a distribution) if the integral over q converges, i.e. if
the integrand is rapidly decreasing in q. To see that this is the case, note that for an
arbitrary but fixed k and large q where here ‘large’ is meant in the Euclidean norm on
R4 , the integrand is vanishing on account of δ(q 2 ) and Θ(k0 − q0 ) as k0 − q0 < 0 for
large q. Loosely speaking, we observe the following: by the form of a convolution,
the Fourier transform of ω2 is multiplied by the same Fourier transform, but with
negative momentum. Since the ω2 has only Fourier support in one ‘energy direction’,
namely the positive one, the intersection of its Fourier support and the same support
evaluated with negative momentum is compact, and the convolution therefore welldefined. Moreover, as this statement only relies on the large momentum behaviour
of Fourier transforms, it holds equally in the case of massive fields, as the mass shell
approaches the light cone for large momenta.
The outcome of the above considerations is the insight that, if we want to define
a sensible generalisation of normal ordering in curved spacetimes, we have to select
states whose two-point functions are singular, but regular enough to allow for pointwise multiplication. Even though general curved spacetimes are not translationally
invariant and therefore do not allow to define a global Fourier transform and a related
global energy positivity condition, one could think that this task can be achieved by
some kind of a ‘local Fourier transform’ and a related ‘local energy positivity condition’ because only the ‘large momentum behaviour’ is relevant. In fact, as showed
in the pioneering work of Radzikowski [103, 104], this heuristic idea can be made
precise in terms of microlocal analysis, a modern branch of Mathematics. Microlocal
analysis gives a rigorous way to define the ‘large momentum behaviour’ of a distribution in a coordinate-independent manner and in the aforementioned works [103,
104], it has been shown that so-called Hadamard states, which have already been
known to allow for a sensible renormalisation of the stress-energy tensor [120, 121,
123], indeed fulfil a local energy positivity condition in the sense that their two-point
function has a specific wave front set. Based on this, Brunetti, Fredenhagen, Köhler,
Hollands, and Wald [18, 19, 70–72] have been able to show that one can as a matter
of fact define normal ordering and perturbative interacting quantum field theories
based on Hadamard states essentially in the same way as on Minkowski spacetimes.
2.4 Hadamard States
41
Though, it turned out that there is a big conceptual difference to flat spacetime quantum field theories, namely, new regularisation freedoms in terms of curvature terms
appear. Although these are finitely many, and therefore lead to the result that theories
which are perturbatively renormalisable in Minkowski spacetime retain this property
in curved spacetimes [19, 70], the appearance of this additional renormalisation freedom may have a profound impact on the backreaction of quantum fields on curved
spacetimes, as we will discuss in the last chapter of this monograph.
As we have already seen at the example of the massless Minkowski vacuum,
Hadamard states can be approached from two angles. One way to discuss them is
to look at the concrete realisation of their two-point function in ‘position space’.
This treatment has lead to the insight that Hadamard states are the sensible starting
point for the definition of a regularised stress-energy tensor [120, 121, 123], and
it is well-suited for actual calculations in particular. On the other hand, the rather
abstract study of Hadamard states based on microlocal analysis is useful in order to
tackle and solve conceptual problems. Following our discussion of the obstructions
in the definition of normal ordering, we shall start our treatment by considering
the microlocal aspects of Hadamard states. A standard monograph on microlocal
analysis is the book of Hörmander [75], who has also contributed a large part to this
field of Mathematics [40, 74]. Introductory treatments can be found in [19, 83, 112,
115].
Let us start be introducing the notion of a wave front set. To motivate it, let us recall
that a smooth function on Rm with compact support has a rapidly decreasing Fourier
transform. If we take an distribution u in Γ0 (Rm ) and multiply it by an f ∈ Γ0 (Rm )
with f (x0 ) = 0, then u f is an element of Γ (Rm ), i.e., a distribution with compact
support. If f u were smooth, then its Fourier transform f u would be smooth and
rapidly decreasing. The failure of f u to be smooth in a neighbourhood of x0 can
therefore be quantitatively described by the set of directions in Fourier space where
f u is not rapidly decreasing. Of course it could happen that we choose f badly and
therefore ‘cut’ some of the singularities of u at x0 . To see the full singularity structure
of u at x0 , we therefore need to consider all test functions which are non-vanishing
at x0 . With this in mind, one first defines the wave front set of distributions on Rm
and then extends it to curved manifolds in a second step.
Definition 2.10 A neighbourhood Γ of k0 ∈ Rm is called conic if k ∈ Γ implies
λk ∈ Γ for all λ ∈ (0, ∞). Let u ∈ Γ0 (Rm ). A point (x0 , k0 ) ∈ Rm × (Rm \ {0}) is
called a regular directed point of u if there is an f ∈ Γ0 (Rm ) with f (x0 ) = 0 such
that, for every n ∈ N, there is a constant Cn ∈ R fulfilling
|
f u(k)| ≤ Cn (1 + |k|)−n
for all k in a conic neighbourhood of k0 . The wave front set WF(u) is the complement
in Rm × (Rm \ {0}) of the set of all regular directed points of u.
We immediately state a few important properties of wave front sets, the proofs of
which can be found in [75] (see also [112]).
42
2 Algebraic Quantum Field Theory on Curved Spacetimes
Theorem 2.6 Let u ∈ Γ0 (Rm ).
(a) If u is smooth, then WF(u) is empty.
(b) Let P be an arbitrary partial differential operator. It holds
WF(Pu) ⊂ WF(u) .
(c) Let U , V ⊂ Rm , let u ∈ Γ0 (V ), and let χ : U → V be a diffeomorphism. The
pull-back χ ∗ (u) of u defined by χ ∗ u( f ) = u(χ∗ f ) for all f ∈ Γ0 (U ) fulfils
.
WF(χ ∗ u) = χ ∗ WF(u) = (χ −1 (x), χ ∗ k) | (x, k) ∈ WF(u) ,
where χ ∗ k denotes the push-forward of χ in the sense of cotangent vectors.
Hence, the wave front set transforms covariantly under diffeomorphisms as an
element of T ∗ Rm , and we can extend its definition to distributions on general
curved manifolds M by patching together wave front sets in different coordinate
patches of M. As a result, for u ∈ Γ0 (M), WF(u) ⊂ T ∗ M \ {0}, where 0 denotes
the zero section of T ∗ M.
(d) Let u 1 , u 2 ∈ Γ0 (M) and let
.
WF(u 1 ) ⊕ WF(u 2 ) = {(x, k1 + k2 ) | (x, k1 ) ∈ WF(u 1 ), (x, k2 ) ∈ WF(u 2 )} .
If WF(u 1 ) ⊕ WF(u 2 ) does not intersect the zero section, then one can define the
product u 1 u 2 in such a way that it yields a well-defined distribution in Γ0 (M)
and that it reduces to the standard pointwise product of smooth functions if u 1
and u 2 are smooth. Moreover, the wave front set of such product is bounded in
the following way
WF(u 1 u 2 ) ⊂ WF(u 1 ) ∪ WF(u 2 ) ∪ (WF(u 1 ) ⊕ WF(u 2 )) .
Note that the wave front set transforms as a subset of the cotangent bundle on
account of the covector nature of k in exp(ikx). The last of the above statements
is exactly the criterion for pointwise multiplication of distributions we have been
looking for. Namely, from (2.8) and (2.7) one can infer that the wave front set of the
Minkowskian two-point function (for m ≥ 0) is [115]
WF(ω2 ) = (x, y, k, −k) ∈ T ∗ M2 | x = y, (x − y)2 = 0, k||(x − y), k0 > 0 (2.9)
∪ (x, x, k, −k) ∈ T ∗ M2 | k 2 = 0, k0 > 0 ,
particularly, it is the condition k0 > 0 which encodes the energy positivity of the
Minkowskian vacuum state. We can now rephrase our observation that the pointwise
square of ω2 (x, y) is a well-defined distribution by noting that WF(ω2 ) ⊕ WF(ω2 )
does not contain the zero section. In contrast, we know that the δ-distribution δ(x) is
2.4 Hadamard States
43
singular at x = 0 and that its Fourier transform is a constant. Hence, its wave front
set reads
WF(δ) = {(0, k) | k ∈ R \ {0}} ,
and we see that the δ-distribution does not have a ‘one-sided’ wave front set and,
hence, can not be squared. The same holds if we view δ as a distribution δ(x, y) on
Γ0 (R2 ). Then
WF(δ(x, y)) = {(x, x, k, −k) | k ∈ R \ {0}} .
The previous discussion suggests that a generalisation of (2.9) to curved spacetimes is the sensible requirement to select states which allow for the construction of
Wick polynomials. We shall now define such a generalisation.
Definition 2.11 Let ω be a state on A (M). We say that ω fulfils the Hadamard
condition and is therefore a Hadamard state if its two-point function ω2 fulfils
WF(ω2 ) = (x, y, k x , −k y ) ∈ T ∗ M 2 \ {0} | (x, k x ) ∼ (y, k y ), k x 0 .
Here, (x, k x ) ∼ (y, k y ) implies that there exists a null geodesic c connecting x to y
such that k x is coparallel and cotangent to c at x and k y is the parallel transport of
k x from x to y along c. Finally, k x 0 means that the covector k x is future-directed.
Having discussed the rather abstract aspect of Hadamard states, let us now turn to
their more concrete realisations. To this avail, let us consider a geodesically convex
set O in (M, g), see Sect. 2.1. By definition, there are open subsets Ox ⊂ Tx M such
that the exponential map expx : Ox → O is well-defined for all x ∈ O, i.e. we can
introduce Riemannian normal coordinates on O. For any two points x, y ∈ O, we
can therefore define the half squared geodesic distance σ (x, y) as
. 1
−1
σ (x, y) = g exp−1
x (y), exp x (y) .
2
This entity is sometimes also called Synge’s world function and is both smooth and
symmetric on O × O. Moreover, one can show that it fulfils the following identity
μ
σ;μ σ; = 2σ ,
(2.10)
where the covariant derivatives are taken with respect to x (even though this does
not matter by the symmetry of σ ), see for instance [57, 102]. Let us introduce a
couple of standard notations related to objects on O × O such as σ . If V and W are
vector bundles over M with typical fibers constituted by the vector spaces V and W
respectively, then we denote by V W the exterior tensor product of V and W .
V W is defined as the vector bundle over M × M with typical fibre V ⊗ W . The
more familiar notion of the tensor product bundle V ⊗ W is obtained by considering
the pull-back bundle of V W with respect to the map M x → (x, x) ∈ M 2 .
44
2 Algebraic Quantum Field Theory on Curved Spacetimes
Typical exterior product bundles are for instance the tangent bundles of Cartesian
products of M, e.g. T ∗ M T ∗ M = T ∗ M 2 . A section of V W is called a bitensor.
We introduce the Synge bracket notation for the coinciding point limits of a bitensor.
Namely, let B be a smooth section of V W . We define
.
[B](x) = lim B(x, y) .
y→x
With this definition, [B] is a section of V ⊗ W . In the following, we shall denote
by unprimed indices tensorial quantities at x, while primed indices denote tensorial
quantities at y. As an example, let us state the well-known Synge rule, proved for
instance in [23, 102].
Lemma 2.4 Let B be an arbitrary smooth bitensor. Its covariant derivatives at x
and y are related by Synge’s rule. Namely,
[B;μ ] = [B] μ − [B;μ ] .
Particularly, let V be a vector bundle, let f a be a local frame of V defined on O ⊂ M
and let x, y ∈ O. If B is symmetric, i.e. the coefficients Bab (x, y) of
.
B(x, y) = B ab (x, y) f a (x) ⊗ f b (y)
fulfil
B ab (x, y) = B b a (y, x) ,
then
[B;μ ] = [B;μ ] =
1
[B];μ .
2
The half squared geodesic distance is a prototype of a class of bitensors of which
we shall encounter many in the following. Namely, σ fulfils a partial differential
equation (2.10) which relates its higher order derivatives to lower order ones. Hence,
given the initial conditions
[σ ] = 0 ,
[σ;μ ] = 0 ,
[σ;μν ] = gμν
which follow from the very definition of σ , one can compute the coinciding point
limits of its higher derivatives by means of an inductive procedure, see for instance
[23, 37, 56, 102]. As an example, in the case of [σ;μνρ ], one differentiates (2.10)
three times and then takes the coinciding point limit. Together with the already known
relations, one obtains
[σ;μνρ ] = 0 .
2.4 Hadamard States
45
At a level of fourth derivative, the same procedure yields a linear combination of three
coinciding fourth derivatives, though with different index orders. To relate those, one
has to commute derivatives to rearrange the indices in the looked-for fashion, and
this ultimately leads to the appearance of Riemann curvature tensors and therefore to
1
[σ;μνρτ ] = − (Rμρντ + Rμτ νρ ) .
3
A different bitensor of the abovementioned kind we shall need in the following
μ
is the bitensor of parallel transport gρ (x, y). Namely, given a geodesically convex
set O, x, y ∈ O, and a vector v = vμ ∂μ in Ty M, the parallel transport of v from y
to x along the unique geodesic in O connecting x and y is given by the vector ṽ in
Tx M with components
μ ṽμ = gρ vρ .
This definition of the bitensor of parallel transport entails
μ
[gρ ] = δρμ ,
μ
gρ ;α σ; α = 0 ,
μ
gρ σ;
ρ
μ
= −σ; .
In fact, the first two identities can be taken as the defining partial differential equaμ
tion of gρ and its initial condition (one can even show that the mentioned partial
differential equation is an ordinary one). Out of these, one can obtain by the inductive
procedure outlined above
μ
[gρ ;α ] = 0 ,
μ
[gρ ;αβ ] =
1 μ
R
.
2 ναβ
With these preparations at hand, let us now provide the explicit form of Hadamard
states.
Definition 2.12 Let ω2 be the two-point function of a state on A (M), let t be a time
function on (M, g), let
.
σε (x, y) = σ (x, y) + 2iε(t (x) − t (y)) + ε2 ,
and let λ be an arbitrary length scale. We say that ω2 if of local Hadamard form if,
for every x0 ∈ M there exists a geodesically convex neighbourhood O of x0 such
that ω2 (x, y) on O × O is of the form
u(x, y)
σε (x, y)
1
+
w(x,
y)
+
v(x,
y)
log
ε↓0 8π 2 σε (x, y)
λ2
1
.
= lim
(h ε (x, y) + w(x, y)) .
ε↓0 8π 2
ω2 (x, y) = lim
46
2 Algebraic Quantum Field Theory on Curved Spacetimes
Here, the Hadamard coefficients u, v, and w are smooth, real-valued biscalars, where
v is given by a series expansion in σ as
v=
∞
vn σ n
n=0
with smooth biscalar coefficients vn . The bidistribution h ε shall be called Hadamard
parametrix, indicating that it solves the Klein-Gordon equation up to smooth terms.
Note that the above series expansion of v does not necessarily converge on general
smooth spacetimes, however, it is known to converge on analytic spacetimes [58].
One therefore often truncates the series at a finite order n and asks for the w coefficient
to be only of regularity C n , see [78]. Moreover, the local Hadamard form is special
case of the global Hadamard form defined for the first time in [78]. The definition of
the global Hadamard form in [78] assures that there are no (spacelike) singularities
in addition to the lightlike ones visible in the local form and, moreover, that the
whole concept is independent of the chosen time function t. However, as proven by
Radzikowski in [104] employing the microlocal version of the Hadamard condition,
the local Hadamard form already implies the global Hadamard form on account of
the fact that ω2 must be positive, have the causal propagator E as its antisymmetric
part, and fulfil the Klein-Gordon equation in both arguments. It is exactly this last
fact which serves to determine the Hadamard coefficients u, v, and w by a recursive
procedure.
To see this, let us omit the subscript ε and the scale λ in the following, since they
do not influence the result of the following calculations, and let us denote by Px the
Klein-Gordon operator P = − + ξ R + m 2 acting on the x-variable. Applying
Px to h, we obtain potentially singular terms proportional to σ −n for n = 1, 2, 3
and to log σ , as well as smooth terms proportional to positive powers of σ . We
know, however, that the total result is smooth because Px (h + w) = 0 since ω2
is a bisolution of the Klein-Gordon equation and w is smooth. Consequently, the
potentially singular terms in Px h have to vanish identically at each order in σ and
log σ . This immediately implies
Px v = 0 .
(2.11)
and, further, the following recursion relations
− Px u + 2v0;μ σ μ + (x σ − 2)v0 = 0 ,
2u ;μ σ μ + (x σ − 4)u = 0 ,
(2.12)
(2.13)
μ
− Px vn + 2(n + 1)vn+1;μ σ; + (n + 1) (x σ + 2n) vn+1 = 0 , ∀n ≥ 0 .
(2.14)
To solve these recursive partial differential equations, let us now focus on (2.13).
Since the only derivative appearing in this equation is the derivative along the
2.4 Hadamard States
47
geodesic connecting x and y, (2.13) is in fact an ordinary differential equation with
respect to the affine parameter of the mentioned geodesic. u is therefore uniquely
determined once a suitable initial condition is given. Comparing the Hadamard form
with the Minkowskian two-point function (2.7), the initial condition is usually chosen as
[u] = 1 ,
which leads to the well-known result that u is given by the square root of the so-called
Van Vleck-Morette determinant, see for instance [23, 37, 56, 102]. Similarly, given u,
the differential equation (2.12) is again an ordinary one with respect to the geodesic
affine parameter, and it can be immediately integrated since taking the coinciding
point limit of (2.12) and inserting the properties of σ yields the initial condition
[v0 ] =
1
[Px u] .
2
It is clear how this procedure can be iterated to obtain solutions for all vn . Particularly,
one obtains the initial conditions
[vn+1 ] =
1
[Px vn ]
2(n + 1)(n + 2)
for all n > 0. Moreover, one finds that u depends only on the local geometry of
the spacetime, while the vn and, hence, v depend only on the local geometry and the
parameters appearing in the Klein-Gordon operator P, namely, the mass m and the
coupling to the scalar curvature ξ . These observations entail that the state dependence
of ω2 is encoded in the smooth biscalar w, which furthermore has to be symmetric
because it is bound to vanish in the difference of two-point functions yielding the
antisymmetric causal propagator E, viz.
ω2 (x, y) − ω2 (x, y) = ω2 (x, y) − ω2 (y, x) = i E(x, y) .
More precisely, this observation ensues from the following important result obtained
in [89, 90].
Theorem 2.7 The Hadamard coefficients vn are symmetric biscalars.
This theorem proves the folklore knowledge that the causal propagator E is locally
given by
1
i E = lim
(h ε − h −ε ) .
ε↓0 8π 2
Even though we can in principle obtain the vn as unique solutions of ordinary
differential equations, we shall only need their coinciding point limits and coinciding
points limit of their derivatives in what follows. In this respect, the symmetry of the
vn will prove very valuable in combination with Lemma 2.4. In fact, employing the
Hadamard recursion relations, we find the following results [91].
48
2 Algebraic Quantum Field Theory on Curved Spacetimes
Lemma 2.5 The following identities hold for the Hadamard parametrix h(x, y)
[Px h] = [Py h] = −6[v1 ] , [(Px h);μ ] = [(Py h);μ ] = −4[v1 ];μ ,
[(Px h);μ ] = [(Py h);μ ] = −2[v1 ];μ .
It is remarkable that these rather simple computations will be essentially sufficient
for the construction of a conserved stress-energy tensor of a free scalar quantum field
[91]. Particularly, the knowledge of the explicit form of, say, [v1 ] is not necessary
to accomplish such a task. However, if one is interested in computing the actual
backreaction of a scalar field on curved spacetimes, one needs the explicit form of
[v1 ]. One can compute this straightforwardly by the inductive procedure already
mentioned at several occasions and the result is [36, 91]
(1 − 5ξ )R
m4
(6ξ − 1) m 2 R
(6ξ − 1) R 2
+
+
+
8
24
288
120
Rαβγ δ R αβγ δ
Rαβ R αβ
+
−
720
720
m4
(1 − 5ξ )R
(6ξ − 1) m 2 R
(6ξ − 1) R 2
=
+
+
+
8
24
288
120
R2
αβγ
δ
αβ
Cαβγ δ C
+ Rαβ R − 3
+
.
720
[v1 ] =
(2.15)
The Hadamard coefficients are related to the so-called DeWitt-Schwinger coefficients, see for instance [35, 89, 90], which stem from an a priori completely different
expansion of two-point functions. The latter have been computed for the first time
in [23, 24] and can also be found in many other places like, e.g. [35, 56].
Having discussed the Hadamard form to a large extent, let us state the already
anticipated equivalence result obtained by Radzikowski in [103]. See also [107] for
a slightly different proof, which closes a gap in the proof of [103].
Theorem 2.8 Let ω2 be the two-point function of a state on A (M). ω2 fulfils the
Hadamard condition of Definition 2.11 if and only if it is of global Hadamard form.
By the result of [104], that a state which is locally of Hadamard form is already
of global Hadamard form, we can safely replace ‘global’ by ‘local’ in the above
theorem. Moreover, from the above discussion it should be clear that the two-point
functions of two Hadamard states differ by a smooth and symmetric biscalar.
In past works on (algebraic) quantum field theory in curved spacetimes, one
has often considered only on quasifree Hadamard states. For non-quasifree states,
a more general microlocal spectrum condition has been proposed in [18] which
requires certain wave front set properties of the higher order n-point functions of a
non-quasifree state. However, as shown in [108, 109], the Hadamard condition of the
two-point function of a non-quasifree state alone already determines the singularity
2.4 Hadamard States
49
structure of all higher order n-point functions by the CCR. It is therefore sufficient to
specify the singularity structure of ω2 also in the case of non-quasifree states. Note
however, that certain technical results on the structure of Hadamard states have up to
now only been proven for the quasifree case [117].
We close the general discussion of Hadamard states by providing examples and
non-examples.
Examples of Hadamard states
1. Given a Hadamard state ω on A (M), any ‘finite excitation of ω’ is again
Hadamard, i.e. for all A ∈ A (M), ω A defined for all B ∈ A (M) by
.
ω A (B) = ω(A∗ B A)/ω(A∗ A) is Hadamard [109].
2. All vacuum states and KMS (thermal equilibrium) states on ultrastatic spacetimes (i.e. spacetimes with a metric ds 2 = −dt 2 + h i j d x i d x j , with h i j not
depending on time) are Hadamard states [55, 106].
3. Based on the previous statement, it has been proven in in [55] that Hadamard
states exist on any globally hyperbolic spacetime by means of a spacetime deformation argument.
4. The Bunch-Davies state on de Sitter spacetime is a Hadamard state [2, 30, 31].
It has been shown in [30, 31] that this result can be generalised to asymptotically
de Sitter spacetimes, where distinguished Hadamard states can be constructed by
means of a holographic argument; these states are generalisations of the BunchDavies state in the sense that the aforementioned holographic construction yields
the Bunch-Davies state in de Sitter spacetime.
5. Similar holographic arguments have been used in [28, 34, 92, 93] to construct
distinguished Hadamard states on asymptotically flat spacetimes, to rigorously
construct the Unruh state in Schwarzschild spacetimes and to prove that it is
Hadamard in [29], to construct asymptotic vacuum and thermal equilibrium
states in certain classes of Friedmann-Robertson-Walker spacetimes in-see [27]
and the next chapter—and to construct Hadamard states in bounded regions of
any globally hyperbolic spacetime in [32].
6. A interesting class of Hadamard states in general Friedmann-Robertson-Walker
are the states of low energy constructed in [95]. These states minimise the energy
density integrated in time with a compactly supported weight function and thus
loosely speaking minimise the energy in the time interval specified by the support
of the weight function. This construction has been generalised to encompass
almost equilibrium states in [84] and to expanding spacetimes with less symmetry
in [114]. We shall review the construction of these states in the next chapter.
7. A construction of Hadamard states which is loosely related to states of low
energy has been given in [15]. There the authors consider for a given spacetime
(M, g) with the spacetime (N , g) where N is a finite-time slab of M. Given
a smooth timelike-compact function f on M which is identically 1 on N , one
.
considers A = i f E f which can be shown to be a bounded and self-adjoint
operator on L 2 (N , dg x). The positive part A+ of A constructed with standard
functional calculus
defines a two-point function of a quasifree state ω on A (N )
. via ω2 ( f, g) = f, A+ g which can be shown to be Hadamard (at least on
50
2 Algebraic Quantum Field Theory on Curved Spacetimes
classes of spacetimes) [15]. Taking for f the characteristic function of N gives
the Sorkin-Johnston states proposed in [1] which are in general not Hadamard
[45].
8. Hadamard states which possess an approximate local thermal interpretation have
been constructed in [111], see [119] for a review.
9. Given a Hadamard state ω on the algebra A (M) and a smooth solution Ψ of
the field equation PΨ = 0, one can construct a coherent state by redefining
the quantum field φ(x) as φ(x) → φ(x) + Ψ (x)I. The thus induced coherent
state has the two-point function ωΨ,2 (x, y) = ω2 (x, y) + Ψ (x)Ψ (y), which is
Hadamard since Ψ (x) is smooth.
10. A construction of Hadamard states via pseudodifferential calculus was developed
in [59].
Non-examples of Hadamard states
1. The so-called α-vacua in de Sitter spacetime [2] violate the Hadamard condition
as shown in [20].
2. As already mentioned the Sorkin-Johnston states proposed in [1] are in general
not Hadamard [45].
3. A class of states related to Hadamard states, but in general not Hadamard, is
constituted by adiabatic states. These have been introduced in [96] and put on
rigorous grounds by [86]. Effectively, they are states which approximate ground
states if the curvature of the background spacetime is only slowly varying. In
[77], the concept of adiabatic states has been generalised to arbitrary curved
spacetimes. There, it has also been displayed in a quantitative way how adiabatic
states are related to Hadamard states. Namely, an adiabatic state of a specific
order n has a certain Sobolev wave front set (in contrast to the C ∞ wave front set
introduced above) and hence, loosely speaking, it differs from a Hadamard state
by a biscalar of finite regularity C n . In this sense, Hadamard states are adiabatic
states of ‘infinite order’. We will review the concept of adiabatic states in the next
chapter.
Finally, let us remark that one can define the Hadamard form also in spacetimes with
dimensions differing from 4, see for instance [91, 107]. Moreover, the proof of the
equivalence of the concrete Hadamard form and the microlocal Hadamard condition
also holds in arbitrary spacetime dimensions, as shown in [107]. A recent detailed
exposition of Hadamard states may be found in [81].
2.5 Locality and General Covariance
And important aspect of QFT on curved spacetimes is the backreaction of quantum
fields in curved spacetimes, i.e. the effect of quantum matter-energy on the curvature
of spacetime. This of course necessitates the ability to define quantum field theory on
a curved spacetime without knowing the curved spacetime beforehand. It is therefore
2.5 Locality and General Covariance
51
advisable to employ only generic properties of spacetimes in the construction of
quantum fields. This entails that we have to formulate a quantum field theory in a
local way, i.e. only employing local properties of the underlying curved manifold.
In addition, we would like to take into account the diffeomorphism-invariance of
General Relativity and therefore construct covariant quantum fields. This concept
of a locally covariant quantum field theory goes back to many works, of which the
first one could mention is [38], followed by many others such as [123, Chap. 4.6] and
[70, 118]. Building on these works, the authors of [21] have given the first complete
definition of a locally covariant quantum field theory.
As shown in [21], giving such a definition in precise mathematical terms requires
the language of category theory, a branch of mathematics which basically aims to
unify all mathematical structures into one coherent picture. A category is essentially
a class C of objects denoted by obj(C), with the property that, for each two objects A,
B in C there is (at least) one morphism or arrow φ : A → B relating A and B. The
collection of all such arrows is denoted by homC(A, B). Morphisms relating a chain
of three objects are required to be associative with respect to compositions, and one
demands that each object has an identity morphism id A : A → A which leaves all
morphisms φ : A → B starting from A invariant upon composition, i.e. φ ◦ id A = φ.
An often cited simple example of a category is the category of sets Set. The objects of
Set are sets, while the morphism are maps between sets, the identity morphism of an
object just being the identity map of a set. Given two categories C1 and C2 , a functor
F : C1 → C2 is a map between two categories which maps objects to objects and
morphisms to morphisms such that identity morphisms in C1 are mapped to identity
morphism in C2 and the composition of morphisms is preserved under the mapping.
This paragraph was only a very brief introduction to category theory and we refer
the reader to the standard monograph [87] and to the introduction in [113, Sect. 1.7]
for further details. A locally covariant quantum field theory according to [21] should
be a functor from a category of spacetimes to a category of suitable algebras. The
first step in understanding such a construction if of course the definition of a suitable
category of spacetimes.
We have already explained in the previous sections of this chapter that fourdimensional, oriented and time-oriented, globally hyperbolic spacetimes are the
physically sensible class of spacetimes among all curved Lorentzian manifolds. It
is therefore natural to take them as the objects of a potential category of spacetimes. Regarding the morphisms, one could think of various possibilities to select
them among all possible maps between the spacetimes under consideration. However, to be able to emphasise the local nature of a quantum field theory, we shall
take embeddings between spacetimes. This will allow us to require locality by asking that a quantum field theory on a ‘small’ spacetime can be easily embedded into
a larger spacetime without ‘knowing anything’ about the remainder of the larger
spacetime. Moreover, a sensible quantum field theory will depend on the orientation
and time-orientation and the causal structure of the underlying manifold, we should
therefore only consider embeddings that preserve these structures. To this avail, the
authors in [21] have chosen isometric embeddings with causally convex range (see
Sect. 2.1 regarding an explanation of these notions), but since the causal structure
52
2 Algebraic Quantum Field Theory on Curved Spacetimes
of a spacetime is left invariant by conformal transformations, one could also choose
conformal embeddings, as done in [98]. We will nevertheless follow the choice of
[21], since it will be sufficient for our purposes. Let us now subsume the above
considerations in a definition.
Definition 2.13 The category of spacetimes Man is the category having as its
class of objects obj(Man) the globally hyperbolic, four-dimensional, oriented and
time-oriented spacetimes (M, g). Given two spacetimes (M1 , g1 ) and (M2 , g2 ) in
obj(Man), the considered morphisms homMan ((M1 , g1 ), (M2 , g2 )) are isometric embeddings χ : (M1 , g1 ) → (M2 , g2 ) preserving the orientation and timeorientation and having causally convex range χ (M1 ). Moreover, the identity morphism id(M,g) of a spacetime in obj(Man) is just the identity map of M and the
composition of morphisms is defined as the usual composition of embeddings.
The just defined category is sufficient to discuss locally covariant Bosonic quantum field theories. However, for Fermionic quantum field theories, one needs a category which incorporates spin structures as defined in [108, 118]. At this point we
briefly remark that our usage of the words ‘Boson’ and ‘Fermion’ for integer and
half-integer spin fields respectively is allowed on account of the spin-statistics theorem in curved spacetimes proved in [118], see also [41] for a more recent and general
work.
To introduce the notion of a locally covariant quantum field theory and the related
concept of a locally covariant quantum field, we need a few categories in addition
to the one introduced above. By TAlg we denote the category of unital topological ∗-algebras, where for two A1 , A2 in obj(TAlg), the considered morphisms
homTAlg(A1 , A2 ) are continuous, unit-preserving, injective ∗-homomorphisms. In
addition, we introduce the category Test of test function spaces Γ0 (M) of objects
(M, g) in Man, where here the morphisms homTest(Γ0 (M1 ), Γ0 (M2 )) are pushforwards χ∗ of the isometric embeddings χ : M1 → M2 . In fact, by D we shall
denote the functor between Man and Test which assigns to a spacetime (M, g) in
Man its test function space Γ0 (M) and to a morphism in Man its push-forward.
For reasons of nomenclature, we consider TAlg and Test as subcategories of the
category Top of all topological spaces with morphisms given by continuous maps.
Let us now state the first promised definition.
Definition 2.14 A locally covariant quantum field theory is a (covariant) functor
A between the two categories Man and TAlg. Namely, let us denote by αχ the
mapping A (χ ) of a morphism χ in Man to a morphism in TAlg and by A (M, g)
the mapping of an object in Man to an object in TAlg, see the following diagram.
2.5 Locality and General Covariance
53
Then, the following relations hold for all morphisms χi j ∈ homMan((Mi , gi ),
(M j , g j )):
αid M = idA (M,g) .
αχ23 ◦ αχ12 = αχ23 ◦χ12 ,
A locally covariant quantum field theory is called causal if in all cases where
χi ∈ homMan((Mi , gi ), (M, g)) are such that the sets χ1 (M1 ) and χ2 (M2 ) are
spacelike separated in (M, g),
αχ1 (A (M1 , g1 )), αχ2 (A (M2 , g2 )) = {0}
in the sense that all elements in the two considered algebras are mutually commuting.
Finally, one says that a locally covariant quantum field theory fulfils the time-slice
axiom, if the situation that χ ∈ homMan((M1 , g1 ), (M2 , g2 )) is such that χ (M1 , g1 )
contains a Cauchy surface of (M2 , g2 ) entails
αχ (A (M1 , g1 )) = A (M2 , g2 ) .
The authors of [21] also give the definition of a state space of a locally covariant
quantum field theory and this turns out to be dual to a functor, by the duality relation
between states and algebras. One therefore chooses the notation of covariant functor
for a functor in the strict sense, and calls such a mentioned dual object a contravariant
functor. We stress once more that the term ‘local’ refers to the size of spacetime
regions. A locally covariant quantum field theory is such that it can be constructed on
arbitrarily small (causally convex) spacetime regions without having any information
on the remainder of the spacetime. In more detail, this means that the algebraic
relations of observables in such small region are already fully determined by the
information on this region alone. This follows by application of the above definition
to the special case that (M1 , g1 ) is a causally convex subset of (M2 , g2 ).
As shown in [21], the quantum field theory given by assigning the BorchersUhlmann algebra A (M) of the free Klein-Gordon field to a spacetime (M, g) is a
locally covariant quantum field theory fulfilling causality and the time-slice axiom.
This follows from the fact that the construction of A (M) only employs compactly
supported test functions and the causal propagator E. The latter is uniquely given on
any globally hyperbolic spacetime, particularly, the causal propagator on a causally
convex subset (M1 , g1 ) of a globally hyperbolic spacetime (M2 , g2 ) coincides with
the restriction of the same propagator on (M2 , g2 ) to (M1 , g1 ). Finally, causality
follows by the causal support properties of the causal propagator, and the time-slice
axiom follows by Lemma 2.3.
Let us now discuss the notion of a locally covariant quantum field. These fields are
particular observables in a locally covariant quantum field theory which transform
covariantly, i.e. loosely speaking, as a tensor and are in addition constructed only
out of local geometric data. In categorical terms, this means that they are natural
transformations between the functors D and A . We refer to [87] for the notion of a
natural transformation, however, its meaning in our context should be clear from the
following definition.
54
2 Algebraic Quantum Field Theory on Curved Spacetimes
Definition 2.15 A locally covariant quantum field Φ is a natural transformation
between the functors D and A . Namely, for every object (M, g) in Man there exists
a morphism Φ(M,g) : Γ0 (M, g) → A (M, g) in Top such that, for each morphism
χ ∈ homMan((M1 , g1 ), (M1 , g1 )), the following diagram commutes.
Particularly, this entails that
αχ ◦ Φ(M1 ,g1 ) = Φ(M2 ,g2 ) ◦ χ∗ .
It is easy to see that the Klein-Gordon field φ( f ) is locally covariant. Namely,
the remarks on the local covariance of the quantum field theory given by A (M)
after Definition 2.14 entail that an isometric embedding χ : (M1 , g1 ) → (M2 , g2 )
transforms φ( f ) as
αχ (φ( f )) = φ(χ∗ f ) ,
or, formally,
αχ (φ(x)) = φ (χ (x)) .
Hence, local covariance of the Klein-Gordon field entails that it transforms as a
‘scalar’. While the locality and covariance of the Klein-Gordon field itself are somehow automatic, one has to take care that all extended quantities, like Wick powers
and time-ordered products thereof, maintain these good properties. The prevalent
paradigm in algebraic quantum field theory is that all pointlike observables should
be theoretically modelled by locally covariant quantum fields.
A comprehensive review of further aspects and results of locally covariant quantum field theory may be found in [46].
2.6 The Quantum Stress-Energy Tensor
and the Semiclassical Einstein Equation
The aim of this section is to discuss the semiclassical Einstein equation and the
quantum-stress-energy tensor : Tμν : which is the observable whose expectation value
enters this equation. As argued in the previous section, all pointlike observables such
as the quantum stress-energy tensor should be locally covariant fields in the sense
of Definition 2.15. Rather than discussing local covariance for non-linear pointlike
observables only at the example of : Tμν :, it is instructive to review the construction
of general local and covariant Wick polynomials.
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
55
2.6.1 Local and Covariant Wick Polynomials
The first construction of local and covariant general Wick polynomials was given
in [70] based on ideas already implemented for the stress-energy tensor in [123].
Here we would like to review a variant of the construction of [70] in the spirit of the
functional approach to perturbative QFT on curved spacetimes, termed perturbative
algebraic quantum field theory, cf. [17, 51, 52]. Essentially, this point of view on
local Wick polynomials was already taken in [72]. We review here only the case
of the neutral Klein-Gordon field, however, the functional approach is applicable to
general field theories [51, 52, 105].
In the functional approach to algebraic QFT on curved spacetimes, one considers
observables as functionals on the classical field configurations. Upon quantization,
these functionals are endowed with a particular non-commutative product which
encodes the commutation relations of quantum observables. We have already taken
this point of view in the discussion of the Borchers-Uhlmann algebra A (M) as the
result of quantizing a classical symplectic space constructed in Sect. 2.2. In particular,
the smeared quantum scalar field φ( f ) was considered as the quantization of the linear functional F f : Γ (M) → C, F f (φ) = f, φ with f ∈ Γ0C (M). The new aspect
in the approach we shall review now is to consider a much larger class of functionals
on Γ (M). To this avail, we view Γ (M) as the space of off-shell configurations of
the scalar field, whereas Sol ⊂ Γ (M) is the space of on-shell configurations. For
the purpose of perturbation theory it is more convenient to perform all constructions
off-shell first and to go on-shell only in the end, and we shall follow this route as
well, even though perturbative constructions are not dealt with in this monograph.
To this end, we call a functional F : Γ (M) → C smooth if the nth functional
derivatives
⎛
⎞
n
n
d
.
F ⎝φ +
λ j ψ j ⎠
(2.16)
F (n) (φ), ψ1 ⊗ · · · ⊗ ψn =
dλ1 . . . dλn
j=1
λ1 =···=λn =0
exist for all n and all ψ1 , . . . , ψn ∈ Γ0 (M) and if F (n) (φ) ∈ Γ (M n ), i.e. F (n) (φ) is
a distribution. By definition F (n) (φ) is symmetric and we consider only polynomial
functionals, i.e. F (n) (φ) vanishes for n > N and some N . We define the support of
a functional as
.
supp F = {x ∈ M | ∀ neighbourhoods U of x ∃φ, ψ ∈ Γ (M), supp ψ ⊂ U,
such that F(φ + ψ) = F(φ)} .
(2.17)
which coincides with the union of the supports of F (1) (φ) over all φ ∈ Γ (M). The
relevant space of functionals which encompasses all observables of the free neutral
scalar field is the space of microcausal functionals
56
2 Algebraic Quantum Field Theory on Curved Spacetimes
.
Fμc = {F : Γ (M) → C | F smooth, compactly supported,
n
n
WF F (n) ∩ V + ∪ V − = ∅ ,
(2.18)
where V+/− is a subset of the cotangent space formed by the elements whose covectors are contained in the future/past light cones and V +/− denotes is closure. Fμc
contains two subspaces of importance. On the one hand, it contains the space Floc
of local functionals consisting of sums of functionals of the form
F(φ) =
dg x P[φ]μ1 ...μn (x) f μ1 ...μn (x)
M
where P[φ] ∈ Γ (T ∗ M n ) is a tensor such that P[φ](x) is a (tensor) product of
covariant derivatives of φ at the point x with a total order of n and f ∈ Γ0 (T M n ) is
a test tensor. The prime example of a local functional is a smeared field monomial
.
Fk, f (φ) =
dg x φ(x)k f (x) φ k ( f ) ,
f ∈ Γ0C (M) .
(2.19)
M
One the other hand, Fμc contains the space Freg of regular functionals, i.e. all
microcausal functionals whose functional derivatives are smooth such that F (n) (φ) ∈
Γ0C (M n ) for all φ and all n. A prime example of a regular functional is a functional
of the form
F(φ) =
n
f j, φ ,
f 1 , . . . , f n ∈ Γ0C (M) .
(2.20)
j=1
Linear functionals are the only functionals which are both local and regular.
Given a bidistribution H ∈ Γ0C (M 2 ) which (a) satisfies the Hadamard condition
Definition 2.11, (b) has the antisymmetric part H (x, y)−H (y, x) = i E(x, y) defined
by the causal propagator E and a real symmetric part
. 1
Hsym (x, y) = (H (x, y) + H (y, x)) ,
2
and (c) is a bisolution of the Klein-Gordon equation Px H (x, y) = Py H (x, y) = 0,
we define a product indicated by H on Fμc via
∞
. 1
F (n) (φ), H ⊗n G (n) (φ) ,
(F H G)(φ) =
n!
(2.21)
n=0
i.e. by the sum of all possible mutual contractions of F and G by means of H .
This is just an elegant way to implement Wick’s theorem by an algebraic product as
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
57
we shall explain in the following and may be understood in terms of deformation
quantization, cf. [53] for a review of this aspect. Owing to the Hadamard property
of H and the microlocal properties of microcausal functionals, the H -product is
well-defined on Fμc by [75, Theorem 8.2.13] and one can show that F H G ∈ Fμc
for all F, G ∈ Fμc , i.e. Fμc is closed under H .
It is not necessary to require that H is positive and thus the two-point function
ω2 of a Hadamard state ω on A (M). However, independent of whether or not we
require H to be positive, its real and symmetric part is not uniquely fixed by the
conditions we imposed. Given a different H satisfying these conditions, it follows
.
−H
that d = H − H = Hsym
sym is real, symmetric and smooth and that the product
H is related to H by
F H G = αd (α−d (F) H α−d (H )) ,
(2.22)
where αd : Fμc → Fμc is the ‘contraction exponential operator’
⎛
.
⎜
αd = exp ⎝
⎞
dg x dg y d(x, y)
M2
δ ⎟
δ
⎠.
δφ(x) δφ(y)
(2.23)
The previous discussion implies that, given a H satisfying the above-mentioned
.
conditions, we can define a meaningful off-shell algebra W H0 (M) = (Fμc , H ),
.
0
and a corresponding on-shell algebra W H (M) = W H (M)/I , where I is the ideal
generated by Fμc,sol ⊂ Fμc , the microcausal functionals which vanish on on-shell
configurations φ ∈ Sol ⊂ Γ (M). In fact, we have W H (M) = (Fμc /Fμc,sol , H )
and in the following we shall indicate an equivalence class [F] ∈ Fμc /Fμc,sol by
F for simplicity.
As a further implication of the previous exposition we have that W H (M) and
W H (M) constructed with a different H of the required type are isomorphic via αd :
W H (M) → W H (M) with d = H − H . In this sense, we can consider W H (M) as a
particular representation of an abstract algebra W (M) which is independent of H , cf.
[17]. W H (M) is in fact a ∗-algebra and αd : W H (M) → W H (M) is a ∗-isomorphism
if we define the involution (∗-operation) on Fμc via complex conjugation by
.
F ∗ (φ) = F(φ)
which implies
.
(F H G)∗ = G ∗ H F ∗
by the conditions imposed on H . W H (M) may be endowed with a topology induced
by the so-called Hörmander topology [19, 25], and one can show that all continuous
states on W H (M) are induced by Hadamard states on the Borchers-Uhlmann algebra
A (M), cf. [69] (in combination with [109]). In fact, we shall now explain why
W H (M) may be considered as the ‘algebra of Wick polynomials’ and in particular
as an extension of A (M).
58
2 Algebraic Quantum Field Theory on Curved Spacetimes
.
To this avail, we first consider two linear functionals φ( f i ) = F1, fi (φ), i = 1, 2,
cf. (2.19), i.e. the classical field smeared with f 1 , f 2 ∈ Γ0C (M). The definition of
the H -product (2.21) implies
.
[φ( f 1 ), φ( f 2 )] H = φ( f 1 ) H φ( f 2 ) − φ( f 2 ) H φ( f 1 ) = i E( f 1 , f 2 ) .
This indicates that the product H encodes the correct commutation relations among
.
quantum observables. If we consider instead the quadratic local functionals φ 2 ( f i ) =
F2, fi (φ), i = 1, 2, cf. (2.19), i.e. the pointwise square of the classical field smeared
with f 1 , f 2 ∈ Γ0C (M), we find
φ 2 ( f 1 ) H φ 2 ( f 2 ) = φ 2 ( f 1 )φ 2 ( f 2 )
+ 4 dg x dg y d(x, y) φ(x)φ(y)H (x, y) f 1 (x) f 2 (y)
M2
2
+ 2H ( f 1 , f 2 ) ,
which we may formally write as
φ 2 (x) H φ 2 (y) = φ 2 (x)φ 2 (y) + 4φ(x)φ(y)H (x, y) + 2H 2 (x, y) .
This expression may be compared by the expression obtain via Wick’s theorem
: φ 2 (x) : H : φ 2 (y) : H =: φ 2 (x)φ 2 (y) : H +4 : φ(x)φ(y) : H H (x, y) + 2H 2 (x, y)I ,
if we define : · : H to be the Wick-ordering w.r.t. the symmetric part Hsym of H , e.g.
.
: φ(x)φ(y) : H = φ(x)φ(y) − Hsym (x, y)I ,
: φ 2 (x) : H = lim : φ(x)φ(y) : H .
x→y
Consequently, local functionals F ∈ Floc considered as elements of F ∈ W H (M)
correspond to Wick polynomials Wick-ordered with respect to Hsym , formally one
may write : F : H = α−Hsym (F) with α−Hsym defined as in (2.23). In fact, the algebra
F ∈ W H (M) contains also time-ordered products of Wick polynomials [19, 70, 72]
and the perturbative construction of QFT based on W H (M) implies that ‘tadpoles’
are already removed.
We have anticipated that W H (M) is an extension of the Borchers-Uhlmann algebra
A (M), which contains only products of the quantum field φ at different points.
To
.
see this, we consider the algebra A (M) = α−Hsym (Freg /Freg,sol , H ) , where
.
Freg,sol = Freg ∩ Fμc,sol , (Freg /Freg,sol , H ) is a subalgebra of W H (M) =
(Fμc /Fμc,sol , H ) because H closes on regular functionals and we note that α−Hsym
is well-defined on Freg although Hsym is not smooth. One can check that A (M) is
in fact the algebra A (M) = (Freg /Freg,sol , E ) with the product E given by (2.21)
with H replaced by i E/2, and that A (M) is isomorphic to the Borchers-Uhlmann
algebra A (M) defined in Definition 2.8.
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
59
We have mentioned that Hadamard states ω on A (M) induce meaningful states
on W H (M) and that in fact all reasonable states on W H (M) are of this form. To
explain this in more detail, we consider a Gaussian Hadamard state on ω on A (M)
with two-point function ω2 and the algebra Wω2 (M) constructed by means of ω2 .
Given this, we can define a Gaussian Hadamard state on Wω2 (M) by
.
ω(F) = F(φ = 0) ,
∀ F ∈ Fμc
which corresponds to the fact that Wick polynomials Wick-ordered w.r.t. to ω2 have
vanishing expectation values in the state ω, e.g. ω(: φ 2 (x) :ω2 ) = 0. Note that this
definition implies in particular that
ω φ( f 1 ) ω2 φ( f 2 ) = ω2 ( f 1 , f 2 ) ,
i.e. ω2 is, as required by consistency, the two-point correlation function of ω also in the
functional picture. If we prefer to consider the extended algebra W H (M) constructed
with a different H , then we can define the state ω on W H (M) by a pull-back with
respect to the isomorphism αd : W H (M) → Wω2 (M) with d = ω2 − H and αd
as in (2.23). In other words, ω ◦ αd defines a state on W H (M), and this definition
corresponds to e.g.
ω(: φ 2 (x) : H ) = lim (ω2 (x, y) − H (x, y)) ,
x→y
i.e. to a point-splitting renormalisation of the expectation value of the observable
φ 2 (x).
We recall that observable quantities should be local and covariant fields as discussed in the previous section. However, not all local elements of the algebra W H (M)
satisfy this property, i.e. not all local functionals F ∈ Floc considered as elements of
W H (M) correspond to local and covariant Wick polynomials. In particular the functional φ 2 ( f ) = F2, f (φ) : φ 2 ( f ) : H , cf. (2.19), does not correspond to a local and
covariant Wick-square because H is by assumption a bisolution of the Klein-Gordon
equation and thus : φ 2 ( f ) : H does not only depend on the geometry, i.e. the metric
and its derivatives, in the localisation region of the test function f , but also on the
geometry of the spacetime (M, g) outside of the support of f [70]. This is related to
the observation that quite generally local and covariant Hadamard states do not exist
[44]. Notwithstanding, local and covariant elements of W H (M) do exist and, following [70, 72], they can identified by means of a map W H : Floc → Floc ⊂ W H (M)
which should satisfy a number of conditions:
1. W H commutes with functional derivatives, i.e. (W H (F))(1) = W H (F (1) ) and
with the involution W H (F)∗ = W H (F ∗ ).
2. W H satisfies the Leibniz rule.
3. W H is local and covariant.
4. W H scales almost homogeneously with respect to constant rescalings m → λm
and g → λ−2 g of the mass m and metric g.
60
2 Algebraic Quantum Field Theory on Curved Spacetimes
5. W H depends smoothly or analytically on the metric g, the mass m and the coupling
ξ to the scalar curvature present in the Klein-Gordon equation (2.4).
We refer to [72] for a detailed discussion of these conditions, and only sketch their
meaning and physical motivation. To this avail, we consider the smeared local polynomials φ k ( f ) = Fk, f (φ) defined in (2.19) and identify W H (φ k (x)) by : φ k (x) :
omitting the smearing function for simplicity. The first of the above axioms then
imply that [: φ k (x) :, φ(y)] = im E(x, y) : φ m−1 (x) :, i.e. the Wick-ordered
fields satisfy standard commutation relations. The Leibniz rule further demands that
∇μ : φ k (x) :=: ∇μ (φ k (x)) :, i.e. Wick-ordering commutes with covariant derivatives, and the locality and covariance condition demands that : φ k (x) : is a local and
covariant field in the sense of Definition 2.15. Finally, the scaling condition requires
that under constant rescalings m → λm and g → λ−2 g, : φ k (x) : scales (in four
spacetime dimensions) as : φ k (x) :→ λk : φ k (x) : +O(log λ), which among other
things implies that : φ k (x) : has the correct ‘mass dimension’, and the smoothness/analyticity requirement implies that e.g. : φ 2 (x) : may not contain a term like
e.g. exp(ξ −1 )m −4 R −1 (x)(Rμν (x)R μν (x))2 which would be allowed by the previous
conditions.
It has been demonstrated in [70, 72] (see also [91]) that a prescription of defining
local and covariant Wick polynomials exists, but that this prescription is not unique.
In fact, if we consider (in a geodesically convex neighbourhood) a H of the form
(2.12) with w = 0, i.e. a purely geometric Hadamard parametrix, then Wick-ordering
.
w.r.t. to this H , e.g. : φ k (x) :=: φ k (x) : H = α−Hsym (φ k (x)) satisfies all conditions
reviewed above. However, this prescription is not the only possibility, but one can
consider e.g.
: φ 2 (x) : = : φ 2 (x) : H +α R(x) + βm 2
with arbitrary real and dimensionless constants α and β which are analytic functions
of ξ . These constants parametrise the renormalisation freedom of Wick polynomials,
or, in the context of perturbation theory, the renormalisation freedom inherent in
removing tadpoles. Note that a change of scale λ in (2.12) can be subsumed in this
renormalisation freedom as a particular one-parameter family.
The coefficients parametrising the renormalisation freedom of local Wick polynomials may not be fixed within QFT on curved spacetimes, but have to be determined
in a more general framework or by comparison with experiments. We will comment
further on this point when discussing the renormalisation freedom of the stressenergy tensor in the context of cosmology in the next chapter. Note that, by local
covariance, these coefficients are universal and may be fixed once and for all in
all globally hyperbolic spacetimes (of the same dimension). Admittedly, in view of
the above presentation of locality and general covariance, one might think that this
holds only for spacetimes with isometric subregions (or spacetimes with conformally
related subregions on account of [98]). However, given two spacetimes (M1 , g1 ) and
(M2 , g2 ) with not necessarily isometric subregions, one can employ the deformation
argument of [54] to deform, say, (M1 , g1 ) such that it contains a subregion isometric to
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
61
a subregion of (M2 , g2 ). As the renormalisation freedom is parametrised by constants
multiplying curvature terms or dimensionful constants which maintain their form
under such a deformation, one can require that the mentioned constants are the same
on (M1 , g1 ) and (M2 , g2 ) in a meaningful way.
2.6.2 The Semiclassical Einstein Equation
and Wald’s Axioms
The central equation in describing the influence of quantum fields on the background
spacetime—i.e. their backreaction—is the semiclassical Einstein equation. It reads
G μν (x) = 8π G ω(: Tμν (x) :) ,
(2.24)
where the left hand side is given by the standard Einstein tensor G μν = Rμν − 21 Rgμν ,
G denotes Newton’s gravitational constant, and we have replaced the stress-energy
tensor of classical matter by the expectation value of a suitable Wick polynomial
: Tμν (x) : representing the quantum stress-energy tensor evaluated in a state ω.
Considerable work has been invested in analysing how such equation can be derived
via a suitable semiclassical limit from some potential quantum theory of gravity. We
refer the reader to [48, Sect. II.B] for a review of several arguments and only briefly
mention that a possibility to derive (2.24) is constituted by starting from the sum of
the Einstein-Hilbert action SEH (g) and the matter action Smatter (g, Φ) ,
.
S(g, Φ) = SEH (g) − Smatter (g, Φ) ,
.
SEH (g) =
1
16π G
dg x R =
M
1
16π G
(2.25)
d x | det g| R
M
formally expanding a quantum metric and a quantum matter field around a classical
(background) vacuum solution of Einstein’s equation, and computing the equation
of motion for the expected metric while keeping only ‘tree-level’ (0 ) contributions
of the quantum metric and ‘loop-level’ (1 ) contributions of the quantum matter
field. In this work, we shall not contemplate on whether and in which situation
the above mentioned ‘partial one-loop approximation’ is sensible, but we shall take
the following pragmatic point of view: (2.24) seems to be the simplest possibility
to couple the background curvature to the stress-energy of a quantum field in a
non-trivial way. We shall therefore consider (2.24) as it stands and only discuss for
which quantum states and Wick polynomial definitions of : Tμν (x) : it is a selfconsistent equation. A rigorous proof that solutions of the semiclassical Einstein
equation actually exist in the restricted case of cosmological spacetimes has been
given in [99, 101].
62
2 Algebraic Quantum Field Theory on Curved Spacetimes
We first observe that in (2.24) one equates a ‘sharp’ classical quantity on the left
hand side with a ‘probabilistic’ quantum quantity on the right hand side. The semiclassical Einstein equation can therefore only be sensible if the fluctuations of the
stress-energy tensor : Tμν (x) : in the considered state ω are small. In this respect, we
already know that we should consider ω to be a Hadamard state and : Tμν (x) : to be a
Wick polynomial Wick-ordered by means of a Hadamard bidistribution. Namely, the
discussion in the previous sections tells us that this setup at least assures finite fluctuations of : Tμν (x) : as the pointwise products appearing in the computation of such
fluctuations are well-defined distributions once their Hadamard property is assumed.
In fact, this observation has been the main motivation to consider Hadamard states
in the first place [120, 123]. However, it seems one can a priori not obtain more than
these qualitative observations, and that quantitative statements on the actual size of
the fluctuations can only be made a posteriori once a solution of (2.24) is found. An
extended framework where the left hand side of the semiclassical Einstein equation
is also interpreted stochastically is discussed in [100].
Having agreed to consider only Hadamard states and Wick polynomials constructed by the procedures outlined in the last section, two questions remain. Which
Hadamard state and which Wick polynomial should one choose to compute the right
hand side of (2.24)? The first question can ultimately only be answered by actually
solving the semiclassical Einstein equation. Observe that this actually poses a nontrivial problem as the formulation of the semiclassical Einstein equation in principle
requires to specify a map which assigns to a metric g a Hadamard state ωg , whereas
we know that a covariant assignment of Hadamard states to spacetimes does not
exist [44]. This problem can be partially overcome by defining such a map only on a
particular subset of globally hyperbolic spacetimes as we shall see in Sect. 3.2.2. The
question of which Wick polynomials should be taken as the definition of a quantum
stress-energy tensor is also non-trivial, as Wick-ordering turns out to be ambiguous
in curved spacetimes, see [70, 72] and the last section. We have already pointed
out at several occasions that one should define the Wick polynomial : Tμν (x) : in a
local, and, hence, state-independent way. In the context of the semiclassical Einstein
equation the reason for this is the simple observation that one would like to solve
(2.24) without knowing the spacetime which results from this procedure beforehand,
but a state solves the equation of motion and, hence, already ‘knows’ the full spacetime, thus being a highly non-local object. On account of the above considerations,
we can therefore answer the question for the correct Wick polynomial representing
: Tμν (x) : without having to choose a specific Hadamard state ω beforehand. The
following review of the quantum stress-energy tensor will be limited to the case of
the free neutral Klein-Gordon field. An analysis of the case of Dirac fields from the
perspective of algebraic QFT on curved spacetimes may be found in [26], whereas
the case of interacting scalar fields in treated in [72].
To this avail, let us consider the stress-energy tensor of classical matter fields.
Given a classical action Smatter , the related (Hilbert) stress-energy tensor can be
computed as [49, 122]
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
2
δSmatter
.
Tμν = √
.
| det g| δg μν
For the Klein-Gordon action S(g, φ) =
1
2
63
(2.26)
φ, Pφ we find
Tμν =(1 − 2ξ )φ;μ φ;ν − 2ξ φ;μν φ + ξ G μν φ 2
!
"
1
1 2 2
ρ
φ;ρ φ; − m φ .
+ gμν 2ξ(φ)φ + 2ξ −
2
2
(2.27)
A straightforward computation shows that the classical stress-energy tensor is
covariantly conserved on-shell, i.e.
∇ μ Tμν = −(∇ν φ)Pφ = 0 .
Moreover, a computation of its trace yields
μ
g μν Tμν = (6ξ − 1) φφ + φ;μ φ; − m 2 φ 2 − φ Pφ .
Particularly, we see that in the conformally invariant situation, that is, m = 0 and
ξ = 16 , the classical stress-energy tensor has vanishing trace on-shell. In fact, one
can show that this is a general result, namely, the trace of a classical stress-energy
tensor is vanishing on-shell if and only if the respective field is conformally invariant
[49, Theorem 5.1].
In view of the discussion of local and covariant Wick polynomials in the previous section, it seems natural to define the quantum stress-energy tensor : Tμν (x) :
just as the Wick polynomial : Tμν (x) : H Wick-ordered with respect to a purely geometric Hadamard parametrix H = 8πh 2 of the form (2.12) with w = 0. Given a
Hadamard state ω whose two-point function is (locally) of the form ω2 = H + 8πw 2 ,
the expectation value of : Tμν (x) : H in this state may be computed as
can can (x, y)w(x, y)
Dμν w
Dμν
ω : Tμν (x) : H = lim
=
,
x→y
8π 2
8π 2
can may be obtained from the classical stresswhere the bidifferential operator Dμν
energy tensor as
can
Dμν
(x, y) =
1 δ 2 Tμν
= (1 − 2ξ )gνν ∇μ ∇ν − 2ξ ∇μ ∇ν + ξ G μν
2 δφ(x)δφ(y)
!
"
1
1
gρρ ∇ ρ ∇ρ − m 2 ,
+ gμν 2ξ x + 2ξ −
2
2
and where we recall the Synge bracket notation for coinciding point limits of bitensors, cf. Sect. 2.4. Here, unprimed indices denote covariant derivatives at x, primed
indices denote covariant derivatives at y and gνν is the bitensor of parallel transport,
64
2 Algebraic Quantum Field Theory on Curved Spacetimes
cf. Sect. 2.4. However, as we shall discuss in a bit more detail in the following, this
particular definition of a quantum stress-energy tensor is not satisfactory because it
does not yield a covariantly conserved quantity which is a necessary condition for
the semiclassical Einstein equation to be well-defined.
The first treatment of the quantum stress-energy tensor from an algebraic point of
view was the analysis of Wald in [120]. At the time [120] appeared, workers in the field
had computed the expectation value of the quantum stress-energy tensor by different
renormalisation methods like adiabatic subtraction, dimensional regularisation, ζ function regularisation and DeWitt-Schwinger point-splitting regularisation (see [13]
and also [62, 88]) and differing results had been found. From the rather modern point
of view we have reviewed in the previous sections, it is quite natural and unavoidable
that renormalisation in curved spacetimes is ambiguous. However, the axioms for the
(expectation value of) the quantum stress-energy tensor introduced in [120] helped
to clarify the case at that time and to understand that in principle all employed
renormalisation schemes were correct in physical terms. Consequently, the apparent
differences between them could be understood on clear conceptual grounds. These
axioms (in the updated form presented in [123]) are:
1. Given two (not necessarily Hadamard) states ω and ω such that the difference
of their two-point functions ω2 − ω2 is smooth, ω(: Tμν (x) :) − ω (: Tμν (x) :) is
equal to
can
ω2 (x, y) − ω2 (x, y) .
Dμν
2. ω(: Tμν (x) :) is locally covariant in the following sense: let
χ : (M, g) → (M , g )
be defined as in Sect. 2.5 and let αχ denote the associated continuous, unitpreserving, injective ∗-morphisms between the relevant enlarged (abstract) algebras W (M, g) and W (M , g ). If two states ω on W (M, g) and ω on W (M , g )
respectively are related via ω = ω ◦ αχ , then
ω : Tμ ν (x ) : = χ∗ ω : Tμν (x) : ,
where χ∗ denotes the push-forward of χ in the sense of covariant tensors.
3. Covariant conservation holds, i.e.
∇ μ ω : Tμν (x) : = 0 .
4. In Minkowski spacetime M, and in the relevant Minkowski vacuum state ωM
ωM : Tμν (x) : = 0 .
5. ω(: Tμν (x) :) does not contain derivatives of the metric of order higher than 2.
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
65
Some of these axioms are just special cases of the axioms for local Wick polynomials reviewed in Sect. 2.6.1. In fact, the first of these axioms is just a variant of
the requirement that local Wick polynomials have standard commutation relations.
In the case of : Tμν : this implies that two valid definitions of this observable can only
differ by a term proportional to the identity. The second axiom is just the locality and
covariance of Wick polynomials here formulated on the level of expectation values.
The condition that the quantum-stress energy tensor has vanishing expectation value
on Minkowski spacetime in the corresponding vacuum state is not compatible with
the requirement that local Wick polynomials depend smoothly on the mass m, cf.
[91, Theorem 2.1(e)] and can be omitted because it is not essential. The last axiom is
motivated by the wish to ensure that the solution theory of the semiclassical Einstein
equation does not depart ‘too much’ from the one of the classical Einstein equation.
In particular one would like to have that all solutions of the semiclassical Einstein
equation behave well in the classical limit → 0. Wald himself had realised, however, that this axiom could not be satisfied for massless theories without introducing
an artificial length scale into the theory; therefore, the axiom has been discarded.
Using these axioms as well as a variant of the scaling requirement for local Wick
polynomials, Wald could prove that a uniqueness result for ω(: Tμν :) (and thus
for : Tμν : itself by the first axiom) can be obtained, namely that two valid definitions
: Tμν : and : Tμν : of the quantum stress-energy tensor can only differ by a term of
the form
: Tμν : − : Tμν := α1 m 4 gμν + α2 m 2 G μν + α3 Iμν + α4 Jμν + εK μν , (2.28)
where αi are real and dimensionless constants and the last three tensors appearing
above are the conserved local curvature tensors
1 2
1
δ
.
2
R + 2R + 2R;μν + 2R Rμν ,
dg x R = −gμν
Iμν = √
2
| det g| δg μν
M
1
δ
.
dg x Rαβ R αβ
Jμν = √
| det g| δg μν
M
K μν
1
= − gμν (Rμν R μν + R) + R;μν − Rμν + 2Rαβ R αμβν ,
2
1
δ
.
=√
dg x Rαβγ δ R αβγ δ
| det g| δg μν
(2.29)
M
1
αβγ
α β
= − gμν Rαβγ δ R αβγ δ + 2Rαβγ μ R ν + 4Rαβ R μ ν
2
− 4Rαμ R αν − 4Rμν + 2R;μν .
This uniqueness result follows by using the first of Wald’s axioms in order to observe
that : Tμν : − : Tμν : is a c-number. From the locality and covariance axiom in
combination with conservation it follows that this c-number must be a local and
66
2 Algebraic Quantum Field Theory on Curved Spacetimes
conserved curvature tensor whereas the scaling condition implies that it has the
correct mass dimension. As [47] pointed out, these requirements do not fix : Tμν :
− : Tμν : to be of the above form, but demanding in addition that : Tμν : depends in a
smooth or analytic way on m and g as in [70, 72] is sufficient to rule out the additional
terms mentioned in [47] so that (2.28) indeed classifies the full renormalisation
freedom compatible with the axioms of local Wick polynomials introduced in [70,
72] and the conservation of : Tμν :.
In fact, we will see in the next section that changing the scale λ in the regularising
Hadamard bidistribution amounts to changing : Tμν : exactly by a tensor of this form
and, furthermore, the attempt to renormalise perturbative Einstein-Hilbert quantum
gravity at one loop order automatically yields a renormalisation freedom in form of
such a tensor as well [116].1 Moreover, using the Gauss-Bonnet-Chern theorem in
four dimensions, which states that
dg x Rμνρτ R μνρτ − 4Rμν R μν + R 2
M
is a topological invariant and, therefore, has a vanishing functional derivative with
respect to the metric [3, 116], one can restrict the freedom even further by removing
K μν from the list of allowed local curvature tensors as K μν = 4Jμν − Iμν . Finally,
the above tensors all have a trace proportional to R and thus the linear combination
Iμν − 3Jμν is traceless.
2.6.3 A Conserved Quantum Stress-Energy Tensor
After some dispute about computational mistakes (see the discussion in [121]) it had
soon be realised that the quantum stress-tensor : Tμν : H defined by Wick-ordering the
canonical expression by means of a purely geometric Hadamard bidistribution H is
not conserved although it satisfies the other conditions for local Wick polynomials
mentioned at the end of Sect. 2.6.1. The reason for this is the fact that, in contrast to the
two-point function of a Hadamard state, a purely geometric Hadamard bidistribution
fails to satisfy the equation of motion. Consequently ∇ μ : Tμν : H =: ∇ μ Tμν : H is (in
four spacetime dimensions) the covariant divergence of a non-vanishing and nonconserved local curvature term ∇ μ : Tμν : H = ∇ μ Cμν = 0. The obvious solution to
this problem has been to compute Cμν and then to define a conserved stress-tensor by
.
: Tμν : = : Tμν : H −Cμν I ,
(2.30)
1 In fact, at least in the case of scalar fields, the combination of the local curvature tensors appearing
as the finite renormalisation freedom in [116] is, up to a term which seems to be an artifact of the
dimensional regularisation employed in that paper, the same that one gets via changing the scale in
the regularising Hadamard bidistribution.
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
67
i.e. by just subtracting this conservation anomaly. It was found that this conserved
stress tensor still has a trace anomaly, i.e. the trace of any stress-energy tensor which
satisfies the Wald axioms is non-vanishing in the conformally invariant case m = 0,
ξ = 16 [121].
Two proposals have been made in order to motivate the ad-hoc subtraction (2.30)
on conceptual grounds. In [91] it has been suggested that the Wick-ordering prescription should be kept, but that the classical expression Tμν entering the definition
of : Tμν : H should be modified in such a way that it coincides with the canonical
classical stress-energy tensor on-shell but gives a conserved observable upon quantization. In [72] instead it has been argued that the Wick-ordering prescription should
be modified without changing the classical expression. This point of view has the
advantage that it fits into the general framework of defining local and covariant Wick
polynomials introduced in [70, 72]. In particular it is possible to alter uniformly the
definition of all local Wick polynomials induced by Wick-ordering with respect to
a purely geometric Hadamard parametrix H in such a way that (a) all axioms for
local Wick polynomials mentioned at the end of Sect. 2.6.1 are still satisfied and (b)
the canonical stress-energy tensor Wick-ordered with respect to this prescription is
conserved [72]. In the aforementioned reference it has also been argued that this
point of view has the further advantage that it is applicable even to perturbatively
constructed interacting models. Notwithstanding, we shall review the approach of
[91] in the following for ease of presentation.
To this avail, we modify the classical stress-energy tensor by setting
c .
can
= Tμν
+ cgμν φ Pφ ,
Tμν
can is the canonical expression and c is a suitable constant to be fixed later.
where Tμν
We then define : Tμν : by
.
c
:H ,
: Tμν : = : Tμν
where : · : H indicates Wick-ordering (i.e. point-splitting regularisation) with respect
to a purely geometric H = 8πh 2 of the form (2.12) with w = 0. Following the
arguments of the previous section, the expectation value of : Tμν : in a Hadamard
state ω whose two-point function is (locally) of the form ω2 = H + 8πw 2 may be
computed as
c Dμν w
ω : Tμν (x) : =
,
8π 2
(2.31)
where
c .
can
= Dμν
Dμν
+ cgμν Px
ν
= (1 − 2ξ )gν ∇μ ∇ν − 2ξ ∇μ ∇ν + ξ G μν
!
"
1
1
gρρ ∇ ρ ∇ρ − m 2 + cgμν Px .
+ gμν 2ξ x + 2ξ −
2
2
(2.32)
68
2 Algebraic Quantum Field Theory on Curved Spacetimes
The following result can now be shown [91].
Theorem 2.9 Let ω(: Tμν (x) :) be defined as in (2.31) with c = 13 .
(a) ω(: Tμν (x) :) is covariantly conserved, i.e.
∇ μ ω : Tμν (x) : = 0 .
(b) The trace of ω(: Tμν (x) :) equals
1
1
g μν ω : Tμν (x) : =
[v1 ] −
4π 2
8π 2
=
1
2880π 2
+
1
4π 2
#
1
3
− ξ + m 2 [w]
6
5
R2
(6ξ − 1) R 2 + 6(1 − 5ξ ) R + Cαβγ δ C αβγ δ + Rαβ R αβ −
2
3
m4
(6ξ − 1) m 2 R
+
8
24
−
1
8π 2
$
1
3
− ξ + m 2 [w] ,
6
which, for m = 0 and ξ = 16 , constitutes the trace anomaly of the quantum
stress-energy tensor.
(c) The conservation and trace anomaly are independent of the chosen scale λ in
the Hadamard parametrix h. Namely, a change
λ → λ
results in
ω(: Tμν (x) :) → ω(: Tμν (x) :) = ω(: Tμν (x) :) + δTμν ,
where
. 2 log λ/λ c 2 log λ/λ can δTμν =
Dμν v =
Dμν v
8π 2
8π 2
2 log λ/λ
=
8π 2
#
(2.33)
m 2 (6ξ − 1)G μν
m4
1
(6ξ − 1)2
−
gμν +
(Iμν − 3Jμν ) −
Iμν
12
8
360
144
$
is a conserved tensor which has vanishing trace for m = 0 and ξ = 16 .
Proof 1. Leaving c undetermined and employing Synge’s rule (cf. Lemma 2.4), we
compute
&
c
%
μ
c
8π 2 ∇ μ ω : Tμν (x) : = ∇ μ Dμν
w = (∇ μ + gμ ∇ μ )Dμν
w
2.6 The Quantum Stress-Energy Tensor and the Semiclassical Einstein Equation
69
% &
= − gνν ∇ν Px + c(gνν ∇ν Px + ∇ν Px ) w .
Let us now recall that Px (h + w) = 0 and, hence, Px w = −Px h. Inserting this
and the identities found in Lemma 2.5, we obtain
%
&
8π 2 ∇ μ ω : Tμν (x) : = − −gνν ∇ν Px + c(gνν ∇ν Px + ∇ν Px ) h
= (6c − 2) [v1 ];ν .
This proves the conservation for c = 13 .
2. It is instructive to leave c undetermined also in this case. Employing Synge’s rule
and the results of Lemma 2.5, we find
c
w
8π 2 g μν ω : Tμν (x) : = 8π 2 g μν Dμν
1
− ξ + m 2 [w]
= (4c − 1)[Px w] − 3
6
1
= (4c − 1)6[v1 ] − 3
− ξ + m 2 [w] .
6
Inserting c = 13 yields the wanted result.
3. The proof ensues without explicitly computing δTμν in terms of the stated conserved tensors from the following observation. Namely, a change of scale as considered transforms w by a adding a term 2 log λ/λ v. Hence, our computations
for proving the first two statements entail
%
&
8π 2
μ
ν
ν
Px + c(g ∇ν Px + ∇ν Px ) v ,
∇
δT
=
−g
∇
μν
ν
ν
ν
2 log λ/λ
1
8π 2
μν
2
[v] .
−
ξ
+
m
g
δT
=
−(4c
−
1)[P
v]
−
3
μν
x
2 log λ/λ
6
The former term vanishes because Px v = 0 as discussed in Sect. 2.4, and the
same holds for the latter term if we insert ξ = 16 and m = 0.
The proof the second statement clearly shows that there is a possibility to assure
vanishing trace in the conformally invariant case, but this possibility is not compatible
with conservation. Moreover, we have stated the last result in explicit terms in order
to show how a change of scale in the Hadamard parametrix is compatible with
the renormalisation freedom of the quantum stress-energy tensor. Finally, we stress
that the term added to the canonical stress-energy tensor is compatible with local
covariance because the corresponding change of the quantum stress-energy tensor is
proportional to gμν [v1 ], i.e. a local curvature tensor.
70
2 Algebraic Quantum Field Theory on Curved Spacetimes
We would also like to point out that the above explicit form of the trace anomaly has
also been known before Hadamard point-splitting had been developed. Particularly,
the same result had been obtained by means of so-called DeWitt-Schwinger pointsplitting in [23]. This renormalisation prescription is a priori not rigorously defined
on Lorentzian spacetimes and the Hadamard point-splitting computation in [121] had
therefore been the first rigorous derivation of the trace anomaly of the stress-energy
tensor. However, DeWitt-Schwinger point-splitting can be reformulated on rigorous
grounds, cf. [62].
2.7 Further Reading
The review of algebraic quantum field theory on curved spacetimes in this chapter has
covered aspects of this framework which are relevant for the applications discussed
in the following chapter. Recent reviews which deal with aspects and constructions
not covered in the present chapter, or provide further details, are [52, 73] and [7,
46, 53, 81], which are part of [16]. A historical account of quantum field theory in
curved spacetimes may be found in [124].
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